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Sharon Keeley Ali Palin Alan Rix Glenn Rogers Emma Stevens Ami Snelling Claire Thompson Julie Wakeling Tim Wakeling eading: Judith Curran Buck, Jennifer Heller, Amanda Jones, Rafiq Ladhani, Betsey McCarty and Cathy Podeszwa. eading: PrPrPrPrProofroofroofroofroofreading: eading: eading: phic Design: Caroline Batten, Ru... |
the universal set consists of all real numbers (see Topic 1.1.2), unless otherwise specified. Sets are a really useful way of being able to say whether numbers or variables have something in common. There’s some new notation here, but the math isn’t too hard at all. Elements Elements e Collections of e Collections of ... |
or Null) Set has No Elements The Empty (or Null) Set has No Elements An empty set (or null set) is a set without any elements or members. It’s denoted by Δ or { }. e Contained WWWWWithin Other Sets ithin Other Sets ithin Other Sets e Contained e Contained Subsets ar Subsets ar ithin Other Sets Subsets are Contained ith... |
Œ {1, 2, 3, 5}. Section 1.1 Section 1.1 Section 1.1 — Sets and Expressions Section 1.1 Section 1.1 33333 Check it out: Another way of saying that set A and B are equal is that A is a subset of B......and B is a subset of A. Equality of Sets Equality of Sets Equality of Sets Equality of Sets Equality of Sets Two sets a... |
and B = {all odd numbers less than 10 but greater than 0}. Explain whether A = B. ound Up ound Up RRRRRound Up ound Up ound Up RRRRRound Up ound Up ound Up ound Up ound Up Sets sound a little odd but they’re just a way of grouping together types of numbers or variables. You’ve seen a lot of the stuff in this Topic in ... |
of This Topic is about one set of numbers that is really important. You’ll be referring to the real numbers throughout Algebra I. s Has Subsets s Has Subsets eal Number he Set of R R R R Real Number eal Number he Set of TTTTThe Set of he Set of s Has Subsets eal Numbers Has Subsets s Has Subsets eal Number he Set of I... |
Subsets of e Subsets of o Mor tionals — tionals and Ir A rational number is a number that can be expressed in the form p q, where p is an integer and q is a natural number. Rational Numbers: = all numbers that can be expressed as fractions ⎧ ⎪⎪ ⎨ ⎩⎪⎪, where p q∈, ∈ p q ⎫ ⎪⎪ ⎬ ⎭⎪⎪ For example, 3.5 is a rational number ... |
1 Section 1.1 TTTTTopicopicopicopicopic 1.1.31.1.3 1.1.31.1.3 1.1.3 California Standards: 1.0:1.0:1.0:1.0:1.0: Students identify Students identify Students identify Students identify and Students identify use the arithmetic properties integggggererererers and s and s and inte inte subsets of subsets of subsets of inte ... |
5}, B = {0, 3, 6}, C = {2, 4, 6, 8}, D = {1, 3, 5, 7}. 1. Find A » B. 2. Find C » D. 3. Find A » C. 4. Find B » D. Sets is AnAnAnAnAnything in Both Sets ything in Both Sets ything in Both Sets Sets is Sets is section of section of he Inter TTTTThe Inter he Inter ything in Both Sets section of Sets is he Intersection o... |
, 8, 10, 12} and B = {3, 6, 9, 12, 15}. Find A « B and A » B. Solution Solution Solution Solution Solution A « B is the set of all the elements that appear in both A and B. So A ««««« B = {6, 12}. A » B is the set of all the elements that appear in A or B, or both sets. So A »»»»» B = {2, 3, 4, 6, 8, 9, 10, 12, 15}. In... |
each way of grouping elements. 88888 Section 1.1 Section 1.1 Section 1.1 — Sets and Expressions Section 1.1 Section 1.1 TTTTTopicopicopicopicopic 1.1.41.1.4 1.1.41.1.4 1.1.4 California Standards: 1.1: Students use 1.1: Students use 1.1: Students use 1.1: Students use 1.1: Students use prprprprproper umbersssss to umbe... |
write (5 + 0.08) instead, since the two expressions have the same value — they’re describing the same thing. Guided Practice Calculate the value of each of these numeric expressions: 1. 18.4 + 8.23 2. 37.82 – 11.19 3. 716 ÷ 2 4. 1790 ÷ 5 5. 37 ÷ 37 6. 19284 ÷ 19284 For each of the following, write a numeric expression... |
expression, as it contains an unknown quantity. The variable x represents the unknown quantity — the number of inches to be converted to centimeters. An algebraic expression always contains at least one variable (and very often it contains numbers as well). Example Example Example Example Example 22222 Write an algebr... |
Up ound Up RRRRRound Up ound Up ound Up RRRRRound Up ound Up ound Up ound Up ound Up Don’t worry if you come across an expression that contains variables. Variables are just letters or symbols that represent unknown numbers, and expressions containing variables follow all the same rules as numeric expressions. Section... |
x in the algebraic expression 2x b) the coefficient of v in v + 5 c) the coefficient of y in 7 – 6y d) the coefficient of k in 5 – k e) the coefficient of m in 5(3 – 2m) Solution Solution Solution Solution Solution a) x is multiplied by 2, so the coefficient is 2. b) v is the same as 1v, so the coefficient is 1. c) Th... |
essions aic Expr ting You’ve already seen algebraic expressions — they’re expressions containing variables. But if you know the value of the letter, you can evaluate the expression — that means you calculate its value. For example, the algebraic expression 2.54x contains the variable x. When x = 33, the expression 2.5... |
: Students identify and use the arithmetic use the arithmetic use the arithmetic use the arithmetic use the arithmetic prprprprproper ties of subsets of ties of ties of oper oper operties of ties of oper integers and rational, umbersssss,,,,, umber umber eal n eal n irrational, and rrrrreal n eal number umber eal n inc... |
a This is the reflexive property of equality. For any numbers a, b ΠR: if a = b then b = a This is the symmetric property of equality. For any numbers a, b, c ΠR: if a = b and b = c, then a = c This is the transitive property of equality. Guided Practice Name the property of equality being used in each statement. 1.... |
�Œ R and (a × b) ŒŒŒŒŒ R. Example Example Example Example Example 11111 Use the fact that 10 Œ R and 6 Œ R to explain why 16 Œ R and 60 Œ R. Solution Solution Solution Solution Solution 10 Œ R and 6 Œ R, so you can add the two numbers to produce another number that is a member of R. So 10 + 6 = 16 Œ R. You can also mul... |
.2 Section 1.2 1515151515 TTTTTopicopicopicopicopic 1.2.21.2.2 1.2.21.2.2 1.2.2 California Standards: 2.0:2.0:2.0:2.0:2.0: Students under stand stand Students under Students under Students understand stand stand Students under h operaaaaations as tions as tions as h oper h oper and use suc and use suc tions as and use ... |
WWWhahahahahat t t t t YYYYYou ou ou ou ou AdAdAdAdAdd to Mak d to Mak se is TTTTThe he he he he AdAdAdAdAdditiditiditiditiditivvvvve Ine Ine Ine Ine Invvvvverererererse is se is d to Mak se is Every real number has an additive inverse — “additive inverse” is just a more mathematical term for the “negative” of a number... |
its multiplicative inverse, the result is 1 — the multiplicative identity. However, there’s an important exception: zero has no reciprocal — its reciprocal cannot be defined. That means that you can’t divide by zero. More formally this becomes: For every nonzero real number m, there is a multiplicative inverse written... |
per umbers s s s s to umber ties of n n n n number umber ties of ties of oper oper operties of umber ties of oper demonstrate whether assertions are true or false. What it means for you: You’ll use the number line to show real numbers, and you’ll describe them in terms of absolute values. Key words: real numbers absolu... |
number line.” Is this statement true or false? 2. Identify the corresponding real numbers of points A–E on the number line below. A B C D E –5 –4 –3 –2 –. Draw the graph of 6 on a number line. 4. Draw the graph of –2 on a number line. 1818181818 Section 1.2 Section 1.2 Section 1.2 — The Real Number System Section 1.2 ... |
of the number, ignoring its sign. The absolute value of c is written |c|. More algebraically... = c ⎧ c ⎪⎪⎪⎪ ⎪⎪⎪⎪ c if ⎨ c if 0 − c ⎩ c if > 0 = 0 < 0 The absolute value of a number can never be negative. Section 1.2 Section 1.2 Section 1.2 — The Real Number System Section 1.2 Section 1.2 1919191919 Example Example Ex... |
202020 Section 1.2 Section 1.2 Section 1.2 — The Real Number System Section 1.2 Section 1.2 TTTTTopicopicopicopicopic 1.2.41.2.4 1.2.41.2.4 1.2.4 California Standards: 1.0: Students identify and 1.0: Students identify and 1.0: Students identify and 1.0: Students identify and 1.0: Students identify and use the arithmeti... |
WWWhen hen ou Get he Sum is WWWWWhahahahahat t t t t YYYYYou Get ou Get he Sum is TTTTThe Sum is he Sum is d Number hen ou Get he Sum is When you add two real numbers a and b (find the sum a + b), the result can be positive, zero, or negative, depending on a and b. The sum of any two real numbers a and b is: 1. Positiv... |
t YYYYYou Get ou Get oduct is oduct is he Pr TTTTThe Pr he Pr he Product is ou Multipl hen ou Get oduct is he Pr In a similar way, the sign of the product of any real numbers depends on the signs of the numbers being multiplied (the factors). The rules for the sign of the product of any two real numbers are as follows... |
(–5) + 2 + 6 13. (–3) × (–5) × (–2) State the signs of the following expressions. 14. a3 where a < 0. 15. –3a2b2 where a < 0 and b > 0. 16. a2b3c7 where a > 0, b < 0, and c < 0 ound Up ound Up RRRRRound Up ound Up ound Up RRRRRound Up ound Up ound Up ound Up ound Up You’ve done addition and multiplication a lot in pre... |
h operaaaaations as tions as tions as h oper h oper and use suc and use suc tions as and use such oper and use suc tions as h oper and use suc taking the opposite,,,,, finding finding finding taking the opposite taking the opposite finding taking the opposite finding taking the opposite ocal ocal ecipr ecipr the r the... |
as additions: a – b = a + (–b) b – a = b + (–a) = (–a) + b The two subtractions “a – b” and “b – a” contain different addends when they’re written as additions — so a – b π b – a. Subtracting a negative number is the same as adding a positive number (since subtracting means adding the opposite, and the opposite of a n... |
). Example Example Example Example Example 33333 Show that a ÷ b π b ÷ a for any a π b, a, b π 0. Solution Solution Solution Solution Solution Write both divisions as multiplications: a ÷ b = a × (b–1) b ÷ a = b × (a–1) The two divisions “a ÷ b” and “b ÷ a” contain different factors when they’re written as multiplicati... |
RRRRRound Up ound Up ound Up Check over the part on reciprocals until you’re sure you understand it. Although the notation is tricky, the actual ideas behind it should make sense if you read through it carefully. 2626262626 Section 1.2 Section 1.2 Section 1.2 — The Real Number System Section 1.2 Section 1.2 TTTTTopico... |
the answer 4 + 6 = 10. You might know the addition has to be done first, but somebody else might not. To be really clear which parts of a calculation have to be done first, you can use grouping symbols. Some common grouping symbols are: parentheses ( ), brackets [ ], and braces { }. Example Example Example Example Exa... |
)} + (–14) Check it out See Topic 1.3.1 for more about exponents. Check it out When you’re simplifying expressions within grouping symbols, follow steps 2–4 (see Example 3). te Firststststst te Fir t to Evaluaaluaaluaaluaaluate Fir te Fir t to Ev About WWWWWhahahahahat to Ev t to Ev About About ules ules e Other R TTTT... |
the multiplica ultiplica then do the m then do the m dition dition then do the ad then do the ad dition then do the addition dition then do the ad then do the ad Now there are no grouping symbols left, so you can do the rest of the calculation: = 185 – 11 = 174 tion firststststst tion fir tion fir ultiplica ultiplica ... |
Section 1.2 — The Real Number System Section 1.2 Section 1.2 2929292929 TTTTTopicopicopicopicopic 1.2.71.2.7 1.2.71.2.7 1.2.7 California Standards: 1.0: Students identify and 1.0: Students identify and 1.0: Students identify and 1.0: Students identify and 1.0: Students identify and use the arithmetic use the arithmeti... |
b = b × a For example: 2 + 3 = 3 + 2 — commutative property of addition 2 × 3 = 3 × 2 — commutative property of multiplication oup Numbers s s s s AnAnAnAnAny y y y y WWWWWaaaaayyyyy oup Number oup Number ou Can Gr ws — YYYYYou Can Gr ou Can Gr ws — Associatititititivvvvve Lae Lae Lae Lae Laws — ws — Associa Associa o... |
’re using. For any a, b, c Œ R: a × (b + c) = (a × b) + (a × c) So a factor outside parentheses multiplies every term inside. Example Example Example Example Example 11111 Expand: a) 3(x + y) b) x(3 + y) c) 3[(x + y) + z] Solution Solution Solution Solution Solution a) 3(x + y) = 3x + 3y b) x(3 + y) = x·3 + xy = 3x + x... |
use suc tions as and use such oper and use suc tions as h oper and use suc taking the opposite,,,,, finding taking the opposite taking the opposite taking the opposite taking the opposite the reciprocal, taking a root, and raising to a fractional power. They understand and use the rules of exponents. What it means for... |
= –1 × (2a + 3) = [(–1) × (2a)] + [(–1) × (3)] = [(–1 × 2) × a)] + (–3) = –2a + (–3) = –2a – 3 Guided Practice Find the opposite of each of these expressions. 1. t 2. 3 3. t + 3 4. a – 2 5. –x + 1 6. –y – 4 Anything Multiplied by 0 is 0 Anything Multiplied by 0 is 0 Anything Multiplied by 0 is 0 Anything Multiplied by... |
There are two expressions multiplied together to give zero. Either one or the other must equal 0, so x = 0 or x – 1 = 0 — that is, x = 0 or x = 1. (You need to write “or,” since x can’t be both 0 and 1 at the same time.) Independent Practice Find the opposite. 1. –4 2. –a 3. g + 5 Solve each equation. 6. x + 1 = 0 7. ... |
oms eal Number TTTTThe Rhe Rhe Rhe Rhe Real Number Axioms Axioms eal Number eal Number eal Number Axioms Axioms eal Number Axioms Most of what has been covered in this Section has been about the axioms (or postulates) of the real number system. This Topic gives a summary of the axioms, and shows how you can use them. A... |
soofs ps in Pr ps in Pr ustify Ste ustify Ste Axioms to J Axioms to J Use the Use the ustify Steps in Pr Axioms to Justify Ste Use the Axioms to J ps in Pr ustify Ste Axioms to J Use the Use the These axioms can be used to justify steps in a mathematical proof. Example Example Example Example Example 11111 Show that (x... |
utiutivvvvve lae lae lae lae lawwwww Distrib Distrib = x2 + yx + x(–y) + y(–y) Distrib Distrib Distributiutiutiutiutivvvvve lae lae lae lae lawwwww Distrib Distrib = x2 + xy + x(–1·y) + y(–1·y) CommCommCommCommCommutautautautautatititititivvvvve lae lae lae lae law ofw ofw ofw ofw of ×, ×, ×, ×, ×, and m and mand m ult... |
2 — The Real Number System Section 1.2 Section 1.2 3535353535 Independent Practice State the real number property that justifies each statement below: 1. 3 + 5 is a real number. 2. 7 × 2 is a real number. 3. m + c = c + m for any real numbers m and c. 4. mc = cm for any real numbers m and c. 5. 12 × (7 × 4) = (12 × 7) ... |
’ll sometimes have to state which rules you’re using when you’re doing math problems. 3636363636 Section 1.2 Section 1.2 Section 1.2 — The Real Number System Section 1.2 Section 1.2 TTTTTopicopicopicopicopic 1.3.11.3.1 1.3.11.3.1 1.3.1 Section 1.3 Exponent Lawswswswsws Exponent La Exponent La Exponent La Exponent La Ex... |
ica A power is a multiplication in which all the factors are the same. For example, m2 = m × m and m3 = m × m × m are both powers of m. In this kind of expression, “m” is called the base and the “2” or “3” is called the exponent. Example Example Example Example Example 11111 a) Find the volume of the cube shown. Write ... |
power. For example: (mb)3 = mb × mb × mb = (m × b) × (m × b) × (m × b = m3b3. (ma)b = mab (mb)a = maba ⎛ ⎜⎜⎜ ⎝ m b ⎞ a ⎟⎟⎟ = ⎠ a m a b 5) Using rule 1 above: ma × m0 = ma + 0 = ma. So m0 equals 1. m0 = 1 6) It’s also possible to make sense of a negative exponent: ma × m–a = ma–a = m0 = 1 (using rules 1 and 5 above) So... |
y ( )2 3 2 x y− 2 ( 25. 32 26. An average baseball has a radius, r, of 1.45 inches. V Find the volume, V, of a baseball in cubic inches. ( = 4 3 = 1 2, 27. The kinetic energy of a ball (in joules) is given by E 2 where m is the ball's mass (in kilograms) and v is its velocity (in meters per second). If a ball weighs 1... |
al aising to a fr and r and r stand stand popopopopowwwwwererererer..... TTTTThehehehehey under y under y under stand y understand stand y under and use the rules of and use the rules of and use the rules of and use the rules of and use the rules of xponents..... eeeeexponents xponents xponents xponents What it means f... |
and –r is called the minor square root of p. Guided Practice Complete the following. 1. The radicand of 83 is......... 2. The 6th root of t is written........ in radical notation. 3. 9 ×........ = 9 4. b 2 =........ in radical notation. 1 4040404040 Section 1.3 Section 1.3 Section 1.3 — Exponents, Roots, and Fractions... |
essions Also Ha aic Expressions e Squar Also Ha essions aic Expr You can also take the square root of an algebraic expression. Example Example Example Example Example 22222 Find the square root of ( x +1 2. ) Solution Solution Solution Solution Solution x +1 2 = |x + 1|, so the principal square root is x + 1 and the m... |
finding taking a root,oot,oot,oot,oot, taking a r taking a r the reciprocal, taking a r taking a r actional actional aising to a fr aising to a fr and r and r actional aising to a fractional and raising to a fr actional aising to a fr and r and r stand stand popopopopowwwwwererererer..... TTTTThehehehehey under y unde... |
ica ty of Squar operty of Squar ty of oper s a Multiplica mc = ⋅ m c This means that to make finding a square root easier, you can try to factor the radicand first. Example Example Example Example Example 11111 Find the following: a) 400 b) 8 c ) Solution Solution Solution Solution Solution a) 400 = × 4 100 b) 8 = × 4 ... |
1.3 Section 1.3 Guided Practice Find the following. 11. 12. 13. 14. 15. 16 4 25 9 125 16 50 4 x 2 36 Independent Practice Find the following. 1. 49 4× 2. 25 2m 3. 4. 5. 64 9 121 144 t 2 81 6. 48 2t 7. 72 2x 8. 8 2m m, ≠ 0 16. 200 2x, x π 0 17. 242 2a a, ≠ 0 18 19. 27 2y 10. 11. 12. 13. 14. 15. 300 2t, t π 0 2 x y2 81 ... |
ent Frrrrractions EquiEquiEquiEquiEquivvvvvalent F actions actions alent F alent F actions alent F actions Here’s another Algebra I Topic that you’ve seen in earlier grades. You’ve used fractions a lot before, but in Algebra I you’ll treat them more formally. This Topic goes over stuff on simplifying fractions that sho... |
to 2. 12 16 3. 9 12 3 4. In exercises 5-7, convert the fractions to 5 8. 5. 25 40 6. 15 24 4. 21 28 7. 40 64 8. Show that 3 5 and 9 15 are equivalent fractions. 9. Show that 5 10 and 3 6 are equivalent fractions. actions TTTTToooooooooo actions alent Frrrrractions actions alent F te Equivvvvvalent F alent F te Equi ou... |
factor of each of these pairs. 12. 81 and 90 13. 56 and 77 14. 42 and 54 15. 13 and 19 Simplify these fractions. 12 14 4 12 16. 17. 18. 30 33 19. 9 24 Independent Practice 1. Identify the numerator in the fraction 11 13. 2. Identify the denominator in the fraction 14 15. In exercises 3–5, show how to simplify each fra... |
lying and Multiplying and Multiplying and Multiplying and Multiplying and Multiplying and actions actions viding Frrrrractions viding F viding F DiDiDiDiDividing F actions actions viding F viding Frrrrractions DiDiDiDiDividing F actions actions viding F viding F actions viding F actions You did lots of work on multiply... |
ys Givvvvve Solutions in the Simplest F e Solutions in the Simplest F ys Gi AlAlAlAlAlwwwwwaaaaays Gi ys Gi e Solutions in the Simplest F ys Gi You should always give your answer in its simplest form — so with more complicated examples, factor the numerators and denominators and cancel common factors. It’ll save time ... |
Solution Solution Solution 72 96 ÷ 9 144 72 = ⋅ 96 144 9 72 96 ⋅ 144 9 = ⋅ 8 9 ⋅ 12 8 ⋅ ⋅ 12 12 9 ocal ocal ecipr ecipr y the r y the r tion b tion b ultiplica ultiplica write as a m write as a m RRRRReeeeewrite as a m ocal eciprocal y the recipr tion by the r ultiplication b write as a multiplica ocal ecipr y the r t... |
�ll look at adding and subtracting fractions, which is a bit tougher. Section 1.3 Section 1.3 Section 1.3 — Exponents, Roots, and Fractions Section 1.3 Section 1.3 5151515151 TTTTTopicopicopicopicopic 1.3.61.3.6 1.3.61.3.6 1.3.6 California Standards: 2.0:2.0:2.0:2.0:2.0: Students under stand stand Students under Studen... |
1 −. 7 Solution Solution Solution Solution Solution a) Add the numerators and divide the answer by the common denominator, 7) Subtract the second numerator from the first and divide the answer by the common denominator, 7 Guided Practice Perform the indicated operations and simplify each expression in exercises 1–9. 1... |
12 to an equi to an equi alent fr action o er 12 8 12 7 + = 12 15 12 = 15 12 ⋅ 5 3 ⋅ 4 3 = 5 4 AdAdAdAdAdd frd frd frd frd fractions with the same denomina actions with the same denomina actions with the same denomina actions with the same denominatortortortortorsssss actions with the same denomina FFFFFactor the n ac... |
Find the least common multiple of each pair of numbers. 11. 5 and 6 10. 4 and 6 14. 21 and 49 13. 14 and 18 17. 16 and 40 16. 12 and 42 20. 24 and 32 19. 36 and 52 12. 9 and 12 15. 18 and 27 18. 15 and 36 21. 25 and 60 Work out and simplify the following. 22. 5 9 + 8 3 25. 28. 13 18 9 15 + 11 27 − 17 10 23. + 11 12 3 ... |
24.2: Students identify the 24.2: Students identify the 24.2: Students identify the 24.2: Students identify the hhhhhypothesis and conc lusion lusion ypothesis and conc ypothesis and conc lusion ypothesis and conclusion lusion ypothesis and conc in logical deduction. in logical deduction. in logical deduction. in logi... |
. For a formal definition of each, refer to Topic 2.2.1. A lot of Algebra I asks you to give formal proofs for stuff that you covered in earlier grades. You’re sometimes asked to state exactly which property you’re using for every step of a math problem. tical Proofoofoofoofoof tical Pr tical Pr thema thema a Ma h Step... |
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1.4 Section 1.4 Section 1.4 — Mathematical Logic Section 1.4 Section 1.4 5555555555 Guided Practice Complete these statements: 1. A mathematical proof is called a................. each step in a logical way using mathematical................. 2. Mathematical proofs can be written in two columns, with the................. |
, when you solve an equation like the one in Example 2, what you are really saying is: “If 6x + 4 = 22, then the value of x is 3.” A sentence like this can be broken down into two basic parts — a hypothesis and a conclusion. The hypothesis is the part of the sentence that follows “if” — here, it is 6x + 4 = 22. The con... |
x2 + y2 = 16, then x2 = 16 – y2. 6. If d – 12 = 23z, then d = 23z + 12. 7. An animal has four legs if it is a dog. 8. Complete this proof by adding the missing justification steps. x – 7 = 17 (x – 7) + 7 = 17 + 7 [x + (–7)] + 7 = 17 + 7 [x + (–7)] + 7 = 24 x + [(–7) + 7] = 24 x + 0 = 24 x = 24 Given equation............. |
eeeexamples to sho xamples to showwwww xamples to sho xamples to sho counter counter xamples to sho counter counter tion is falsealsealsealsealse thathathathathat an asser tion is f tion is f t an asser t an asser t an assertion is f tion is f t an asser t a singlelelelele t a sing t a sing e tha e tha and recoecoecoec... |
, 16, 25, 36... If you look at the differences between successive terms, you find this: The difference between the first and second terms is 4 – 1 = 3. The difference between the second and third terms is 9 – 4 = 5. The difference between the third and fourth terms is 16 – 9 = 7. The difference between the fourth and f... |
Example 11111 Decide whether the following statement is always true: “2n + 1 is always a prime number, where n is a natural number.” Solution Solution Solution Solution Solution At first, the rule looks believable. If n = 1: 2n + 1 = 21 + 1 = 2 + 1 = 3. This is a prime number, so the rule holds for n = 1. If n = 2: 2n... |
digits is a multiple of 3. Use this information to decide whether 96 is a multiple of 3. Solution Solution Solution Solution Solution The sum of the digits is 9 + 6 = 15, which is divisible by 3. The statement says that a number is a multiple of 3 if the sum of its digits is a multiple of 3. Using deductive reasoning,... |
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Solutions Can Consist of More than One P Solutions Can Consist of Mor e than One P Mor Solutions Can Consist of Solutions Can Consist of Example Example Example Example Example 11111 Find x given that x2 = 9. Solution Solution Solution Solution Solution You can think of the equation as a hypothesis — then you need to ... |
2.3 for more about absolute value equations. Check it out: If |a| < |b|, then: 0 < a < b, or b < a < 0 or a < 0 < b, or b < 0 < a. tement is Nevvvvver er er er er TTTTTrrrrrueueueueue tement is Ne tement is Ne Sometimes a Sta Sometimes a Sta Sometimes a Statement is Ne tement is Ne Sometimes a Sta Sometimes a Sta There... |
— Mathematical Logic Section 1.4 Section 1.4 Example Example Example Example Example 55555 If x is a real number, find the possible values for which the following is true: x2 + 1 < 2x Solution Solution Solution Solution Solution You can use the inequality as a hypothesis. Now you need to find a suitable conclusion usi... |
-11, say whether the algebraic statements are true sometimes, always, or never. 7. 5 + x = 10 10. x2 < 0 8. x2 = –9 11. x2 > x 9. x2 = 49 12. Is this statement true? “If x > 0, then x3 > x2” 13. Find x given that |x| + 7 = 0. 14. What values of x satisfy x2 – 25 = (x + 5)(x – 5)? 15. The area of a rectangle is given by... |
could there be if there were three erasers to collect? How many children could there be if there were four erasers to collect? Look at your answers — what do you notice? The collections can overlap — for example, one child could have erasers a, b, and d, while another could have erasers b, c, and g. Extension 1) If th... |
0: Students simplify 4.0: Students simplify 4.0: Students simplify 4.0: Students simplify eeeeexprxprxprxprxpressions essions essions essions before solving essions linear equations and inequalities in one variable, such as 3(2x – 5) + 4(x – 2) = 12..... What it means for you: You’ll combine like terms to simplify expr... |
Which variable is multiplied by –4 in the algebraic expression 4x2 – 4y + 8 + 4xy? 6. Counting from left to right, which term is the fourth term in the algebraic expression 8x2 + 2xy – 6y + 9xy3 – 4? LikLikLikLikLike e e e e TTTTTerererererms Can Be Combined ms Can Be Combined ms Can Be Combined ms Can Be Combined ms ... |
)x] – 5 = 2x – 5 dition dition ty of ad ad ad ad addition ty of ty of oper oper e pre proper e pre pr Associatititititivvvvve pr Associa Associa dition operty of dition ty of oper Associa Associa dition dition ty of ad ad ad ad addition ty of ty of oper oper CommCommCommCommCommutautautautautatititititivvvvve pre pre p... |
2 × 4c 6868686868 Section 2.1 Section 2.1 Section 2.1 — Algebra Basics Section 2.1 Section 2.1 Independent Practice In Exercises 1–5, determine the number of terms in each algebraic expression: 1. 7b + 14a – 4 3. (27x2 + 4x) – 13 5. 2x + 4xy + 4x2 – (10 + 12y + 19y2) 2. 2a 4. 5 +10x + 20x2 + 3a In Exercises 6–9, simpl... |
) pieces of candy, Leo has (8 – 2x) pieces of candy, and Maria has 8 pieces of candy. Write and simplify an algebraic expression showing the total number of pieces of candy the three friends have to eat. ound Up ound Up RRRRRound Up ound Up ound Up ound Up ound Up RRRRRound Up ound Up ound Up You’ve combined like terms... |
Symbols e Proper ouping Symbols es Gr oper e Pr he Distrib The expression 5(3x + 2) + 2(2x – 1) can be simplified — both parts have an “x” term and a constant term. To simplify an expression like this, you first need to get rid of the grouping symbols. The way to do this is to use the distributive property of multipli... |
a number outside a grouping symbol is negative, like in –7(2x + 1), you have to remember to use the multiplicative property of –1. This means that the signs of the terms within the grouping symbols will change: “+” signs will change to “–” signs and vice versa. Example Example Example Example Example 22222 Simplify th... |
⎜ ⎝ 1 2 + n 2 ⎞ ⎟⎟⎟− ⎠ ⎛ ⎜⎜⎜ 3 ⎝ 1 3 − n 4 ⎞ ⎟⎟⎟ ⎠ 14. Simplify 12(2n – 7) – 9(3 – 4n) + 6(4x – 9). 15. Simplify 5(x – 2) – 7(–4x + 3) – 3(–2x). Section 2.1 Section 2.1 Section 2.1 — Algebra Basics Section 2.1 Section 2.1 7171717171 Independent Practice In Exercises 1–6, simplify the algebraic expressions: 1. –4(a + 2b... |
counting stamps. If x represents Ruby's age, she has 4(x – 4) stamps, Sara has 2(8 – x) stamps, and Keisha has 8(7 + 2x) stamps. Write and simplify an algebraic expression showing the total number of stamps owned by the three friends. ound Up ound Up RRRRRound Up ound Up ound Up RRRRRound Up ound Up ound Up ound Up ou... |
ly y y y y VVVVVariaariaariaariaariabbbbbleslesleslesles Exponents to Multipl Exponents to Multipl ules of ules of Use R Use R ules of Exponents to Multipl Use Rules of Exponents to Multipl ules of Use R Use R To simplify expressions like 4x(x2 – 2x + 1), you need to apply the distributive property as well as rules of ... |
4) 7. 7n(3a + b) – 4a(7n + 2b) 8. 2x(2x2 – x) + x(2x – 8) + 3x(x – 4) 9. 2x(k – 9) – k(x – 7) + xk(4 – 3x) Section 2.1 Section 2.1 Section 2.1 — Algebra Basics Section 2.1 Section 2.1 7373737373 Don’t forget: See Topic 2.2.1 for the difference between expressions and equations. Check it out: Continue simplifying until... |
x + 3) – 3(3x + 2) = 4x + 3. 11. Given b = –1, show that ( 12. Given x = – 5 3, show that 4(2x – 1) – 5(x – 2) = 1. 13. Show whether x = –2 is or is not a solution of –6x – 15 = –17 – 9x. − − 14. If b = –10, show that. 15. Show whether x = – 2 3 is or is not a solution of –6x – 15 = –17 – 9x. 16. Verify that x = 0 is a... |
value of each algebraic expression when the given substitutions are made: 19. 2x(x + 5y) – 3y(y + 3) 20. 6y(yx – 4) – 5(yx – 4) if x = 2, y = 4 if x = 1, y = –1 if a = 0, b = 3, n = 1 7 if x = 4, y = –8 if a = 4, b = 0.2 21. 7n(30 + b) – 4a(7n + 2b) 22. –2y(3yx + 2) 23. –4a(b – 4) 24. 2x(k – a) – k(x – a) + xk(a – 3x)... |
Topicopicopicopicopic 2.2.12.2.1 2.2.12.2.1 2.2.1 Section 2.2 Equality Equality ties of ties of oper oper PrPrPrPrProper Equality ties of Equality operties of Equality ties of oper PrPrPrPrProper Equality Equality ties of ties of oper oper ties of Equality operties of Equality ties of oper Equality California Standards... |
-hand side... 24 – 9 = 15...has the same value as the expression on the right-hand side Some equations contain unknown quantities, or variables. The left-hand side... 2x – 3 = 5...equals the right-hand side The value of x that satisfies the equation is called the solution (or root) of the equation. tions tions action i... |
both sides. If you have a “– 9” that you want to get rid of, you can just add 9 to both sides. In other words, you just need to use the inverse operations. Example Example Example Example Example 22222 Solve x + 10 = 12. Solution Solution Solution Solution Solution x + 10 = 12 x = 12 – 10 Subtr Subtr Subtr act 10 fr a... |
Own vide to Get the y or Di Multipl Multipl As with addition and subtraction, you can get the variable on its own by simply performing the inverse operation. If you have “× 3” on one side of the equation, you can get rid of that value by dividing both sides by 3. If you have a “÷ 3” that you want to get rid of, you ca... |
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