# Chapter 1. Solving Equations and Inequalities 1.1 – Expressions and Formulas.

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Chapter 1

Solving Equations and Inequalities

1.1 – Expressions and Formulas

Order of Operations

1.1 – Expressions and Formulas Order of Operations Parentheses

1.1 – Expressions and Formulas Order of Operations Parentheses Exponents

1.1 – Expressions and Formulas Order of Operations Parentheses Exponents Multiplication

1.1 – Expressions and Formulas Order of Operations Parentheses Exponents Multiplication Division

1.1 – Expressions and Formulas Order of Operations Parentheses Exponents Multiplication Division Addition

1.1 – Expressions and Formulas Order of Operations Parentheses Exponents Multiplication Division Addition Subtraction

1.1 – Expressions and Formulas Order of Operations ParenthesesPlease Exponents Multiplication Division Addition Subtraction

1.1 – Expressions and Formulas Order of Operations ParenthesesPlease ExponentsExcuse Multiplication Division Addition Subtraction

1.1 – Expressions and Formulas Order of Operations ParenthesesPlease ExponentsExcuse MultiplicationMy Division Addition Subtraction

1.1 – Expressions and Formulas Order of Operations ParenthesesPlease ExponentsExcuse MultiplicationMy DivisionDear Addition Subtraction

1.1 – Expressions and Formulas Order of Operations ParenthesesPlease ExponentsExcuse MultiplicationMy DivisionDear AdditionAunt Subtraction

1.1 – Expressions and Formulas Order of Operations ParenthesesPlease ExponentsExcuse MultiplicationMy DivisionDear AdditionAunt SubtractionSally

Example 1

Find the value of [2(10 - 4) 2 + 3] ÷ 5.

Example 1 Find the value of [2(10 - 4) 2 + 3] ÷ 5. [2(10 - 4) 2 + 3] ÷ 5 =

Example 1 Find the value of [2(10 - 4) 2 + 3] ÷ 5. [2(10 - 4) 2 + 3] ÷ 5 = [2(6) 2 + 3] ÷ 5

Example 1 Find the value of [2(10 - 4) 2 + 3] ÷ 5. [2(10 - 4) 2 + 3] ÷ 5 = [2(6) 2 + 3] ÷ 5 [2(36) + 3] ÷ 5

Example 1 Find the value of [2(10 - 4) 2 + 3] ÷ 5. [2(10 - 4) 2 + 3] ÷ 5 = [2(6) 2 + 3] ÷ 5 [2(36) + 3] ÷ 5 [72 + 3] ÷ 5

Example 1 Find the value of [2(10 - 4) 2 + 3] ÷ 5. [2(10 - 4) 2 + 3] ÷ 5 = [2(6) 2 + 3] ÷ 5 [2(36) + 3] ÷ 5 [72 + 3] ÷ 5 75 ÷ 5

Example 1 Find the value of [2(10 - 4) 2 + 3] ÷ 5. [2(10 - 4) 2 + 3] ÷ 5 = [2(6) 2 + 3] ÷ 5 [2(36) + 3] ÷ 5 [72 + 3] ÷ 5 75 ÷ 5 15

Example 2

Evaluate x 2 – y(x + y) if x = 8 and y = 1.5.

Example 2 Evaluate x 2 – y(x + y) if x = 8 and y = 1.5. x 2 – y(x + y) =

Example 2 Evaluate x 2 – y(x + y) if x = 8 and y = 1.5. x 2 – y(x + y) = 8 2 – 1.5(8 + 1.5)

Example 2 Evaluate x 2 – y(x + y) if x = 8 and y = 1.5. x 2 – y(x + y) = 8 2 – 1.5(8 + 1.5) 8 2 – 1.5(8 + 1.5)

Example 2 Evaluate x 2 – y(x + y) if x = 8 and y = 1.5. x 2 – y(x + y) = 8 2 – 1.5(8 + 1.5) 8 2 – 1.5(8 + 1.5) 8 2 – 1.5(9.5)

Example 2 Evaluate x 2 – y(x + y) if x = 8 and y = 1.5. x 2 – y(x + y) = 8 2 – 1.5(8 + 1.5) 8 2 – 1.5(8 + 1.5) 8 2 – 1.5(9.5) 64 – 1.5(9.5)

Example 2 Evaluate x 2 – y(x + y) if x = 8 and y = 1.5. x 2 – y(x + y) = 8 2 – 1.5(8 + 1.5) 8 2 – 1.5(8 + 1.5) 8 2 – 1.5(9.5) 64 – 1.5(9.5) 64 – 14.25

Example 2 Evaluate x 2 – y(x + y) if x = 8 and y = 1.5. x 2 – y(x + y) = 8 2 – 1.5(8 + 1.5) 8 2 – 1.5(8 + 1.5) 8 2 – 1.5(9.5) 64 – 1.5(9.5) 64 – 14.25 49.75

Example 3

Evaluate a 3 + 2bc if a = 2, b = -4, and c = -3. c 2 – 5

Example 3 Evaluate a 3 + 2bc if a = 2, b = -4, and c = -3. c 2 – 5 a 3 + 2bc = c 2 – 5

Example 3 Evaluate a 3 + 2bc if a = 2, b = -4, and c = -3. c 2 – 5 a 3 + 2bc = 2 3 + 2(-4)(-3) c 2 – 5

Example 3 Evaluate a 3 + 2bc if a = 2, b = -4, and c = -3. c 2 – 5 a 3 + 2bc = 2 3 + 2(-4)(-3) c 2 – 5 (-3) 2 – 5

Example 3 Evaluate a 3 + 2bc if a = 2, b = -4, and c = -3. c 2 – 5 a 3 + 2bc = 2 3 + 2(-4)(-3) c 2 – 5 (-3) 2 – 5 = 8 + 2(-4)(-3)

Example 3 Evaluate a 3 + 2bc if a = 2, b = -4, and c = -3. c 2 – 5 a 3 + 2bc = 2 3 + 2(-4)(-3) c 2 – 5 (-3) 2 – 5 = 8 + 2(-4)(-3) 9 – 5

Example 3 Evaluate a 3 + 2bc if a = 2, b = -4, and c = -3. c 2 – 5 a 3 + 2bc = 2 3 + 2(-4)(-3) c 2 – 5 (-3) 2 – 5 = 8 + 2(-4)(-3) 9 – 5 = 8 + 24 9 – 5

Example 3 Evaluate a 3 + 2bc if a = 2, b = -4, and c = -3. c 2 – 5 a 3 + 2bc = 2 3 + 2(-4)(-3) c 2 – 5 (-3) 2 – 5 = 8 + 2(-4)(-3) 9 – 5 = 8 + 24 9 – 5 = 32 4

Example 3 Evaluate a 3 + 2bc if a = 2, b = -4, and c = -3. c 2 – 5 a 3 + 2bc = 2 3 + 2(-4)(-3) c 2 – 5 (-3) 2 – 5 = 8 + 2(-4)(-3) 9 – 5 = 8 + 24 9 – 5 = 32 = 8 4

Example 4

Find the area of the following trapezoid. 16 in. 10 in. 52 in.

Example 4 Find the area of the following trapezoid. 16 in. A = ½h(b 1 + b 2 ) 10 in. 52 in.

Example 4 Find the area of the following trapezoid. 16 in. A = ½h(b 1 + b 2 ) 10 in. 52 in. A = ½h(b 1 + b 2 )

Example 4 Find the area of the following trapezoid. 16 in. A = ½h(b 1 + b 2 ) 10 in. = h 52 in. A = ½h(b 1 + b 2 )

Example 4 Find the area of the following trapezoid. 16 in. = b 1 A = ½h(b 1 + b 2 ) 10 in. = h 52 in. A = ½h(b 1 + b 2 )

Example 4 Find the area of the following trapezoid. 16 in. = b 1 A = ½h(b 1 + b 2 ) 10 in. = h 52 in. = b 2 A = ½h(b 1 + b 2 )

Example 4 Find the area of the following trapezoid. 16 in. A = ½h(b 1 + b 2 ) 10 in. 52 in. A = ½h(b 1 + b 2 ) = ½10(16 + 52)

Example 4 Find the area of the following trapezoid. 16 in. A = ½h(b 1 + b 2 ) 10 in. 52 in. A = ½h(b 1 + b 2 ) = ½10(16 + 52) = ½10(68)

Example 4 Find the area of the following trapezoid. 16 in. A = ½h(b 1 + b 2 ) 10 in. 52 in. A = ½h(b 1 + b 2 ) = ½10(16 + 52) = ½10(68) = 5(68)

Example 4 Find the area of the following trapezoid. 16 in. A = ½h(b 1 + b2) 10 in. 52 in. A = ½h(b 1 + b 2 ) = ½10(16 + 52) = ½10(68) = 5(68) = 340

1.2 – Properties of Real Numbers

Real Numbers

1.2 – Properties of Real Numbers Real Numbers (R)

1.2 – Properties of Real Numbers Real Numbers (R)

1.2 – Properties of Real Numbers Real Numbers (R) Rational

1.2 – Properties of Real Numbers Real Numbers (R) Rational (⅓)

1.2 – Properties of Real Numbers Real Numbers (R) (Q) Rational (⅓)

1.2 – Properties of Real Numbers Real Numbers (R) (Q) Rational (⅓)

1.2 – Properties of Real Numbers Real Numbers (R) (Q) Rational (⅓) Integers

1.2 – Properties of Real Numbers Real Numbers (R) (Q) Rational (⅓) Integers (-6)

1.2 – Properties of Real Numbers Real Numbers (R) (Q) Rational (⅓) (Z) Integers (-6)

1.2 – Properties of Real Numbers Real Numbers (R) (Q) Rational (⅓) (Z) Integers (-6)

1.2 – Properties of Real Numbers Real Numbers (R) (Q) Rational (⅓) (Z) Integers (-6) Whole #’s

1.2 – Properties of Real Numbers Real Numbers (R) (Q) Rational (⅓) (Z) Integers (-6) Whole #’s (0)

1.2 – Properties of Real Numbers Real Numbers (R) (Q) Rational (⅓) (Z) Integers (-6) (W) Whole #’s (0)

1.2 – Properties of Real Numbers Real Numbers (R) (Q) Rational (⅓) (Z) Integers (-6) (W) Whole #’s (0)

1.2 – Properties of Real Numbers Real Numbers (R) (Q) Rational (⅓) (Z) Integers (-6) (W) Whole #’s (0) Natural #’s

1.2 – Properties of Real Numbers Real Numbers (R) (Q) Rational (⅓) (Z) Integers (-6) (W) Whole #’s (0) Natural #’s (7)

1.2 – Properties of Real Numbers Real Numbers (R) (Q) Rational (⅓) (Z) Integers (-6) (W) Whole #’s (0) (N) Natural #’s (7)

1.2 – Properties of Real Numbers Real Numbers (R) (Q) Rational (⅓) (Z) Integers (-6) (W) Whole #’s (0) (N) Natural #’s (1)

1.2 – Properties of Real Numbers Real Numbers (R) (Q) Rational (⅓) Irrational (Z) Integers (-6) (W) Whole #’s (0) (N) Natural #’s (1)

1.2 – Properties of Real Numbers Real Numbers (R) (Q) Rational (⅓) Irrational √ 5 (Z) Integers (-6) (W) Whole #’s (0) (N) Natural #’s (1)

1.2 – Properties of Real Numbers Real Numbers (R) (Q) Rational (⅓) (I) Irrational √ 5 (Z) Integers (-6) (W) Whole #’s (0) (N) Natural #’s (1)

Real Rational Irrational Integers Whole Natural

Example 1

Name the sets of numbers to which each apply.

Example 1 Name the sets of numbers to which each apply.

Example 1 Name the sets of numbers to which each apply.

Example 1 Name the sets of numbers to which each apply. (a) √ 16

Example 1 Name the sets of numbers to which each apply. (a) √ 16 = 4

Example 1 Name the sets of numbers to which each apply. (a) √ 16 = 4 - N

Example 1 Name the sets of numbers to which each apply. (a) √ 16 = 4 - N, W

Example 1 Name the sets of numbers to which each apply. (a) √ 16 = 4 - N, W, Z

Example 1 Name the sets of numbers to which each apply. (a) √ 16 = 4 - N, W, Z, Q

Example 1 Name the sets of numbers to which each apply. (a) √ 16 = 4 - N, W, Z, Q, R

Example 1 Name the sets of numbers to which each apply. (a)√ 16 = 4 - N, W, Z, Q, R (b)-185

Example 1 Name the sets of numbers to which each apply. (a)√ 16 = 4 - N, W, Z, Q, R (b)-185 - Z

Example 1 Name the sets of numbers to which each apply. (a)√ 16 = 4 - N, W, Z, Q, R (b)-185 - Z, Q

Example 1 Name the sets of numbers to which each apply. (a)√ 16 = 4 - N, W, Z, Q, R (b)-185 - Z, Q, R

Example 1 Name the sets of numbers to which each apply. (a)√ 16 = 4 - N, W, Z, Q, R (b)-185 - Z, Q, R (c)√ 20

Example 1 Name the sets of numbers to which each apply. (a)√ 16 = 4 - N, W, Z, Q, R (b)-185 - Z, Q, R (c)√ 20 - I, R

Example 1 Name the sets of numbers to which each apply. (a)√ 16 = 4 - N, W, Z, Q, R (b)-185 - Z, Q, R (c)√ 20 - I, R (d) -⅞

Example 1 Name the sets of numbers to which each apply. (a)√ 16 = 4 - N, W, Z, Q, R (b)-185 - Z, Q, R (c)√ 20 - I, R (d) -⅞ - Q

Example 1 Name the sets of numbers to which each apply. (a)√ 16 = 4 - N, W, Z, Q, R (b)-185 - Z, Q, R (c)√ 20 - I, R (d) -⅞ - Q, R

Example 1 Name the sets of numbers to which each apply. (a)√ 16 = 4 - N, W, Z, Q, R (b)-185 - Z, Q, R (c)√ 20 - I, R (d) -⅞ - Q, R __ (e) 0.45

Example 1 Name the sets of numbers to which each apply. (a)√ 16 = 4 - N, W, Z, Q, R (b)-185 - Z, Q, R (c)√ 20 - I, R (d) -⅞ - Q, R __ (e) 0.45 - Q

Example 1 Name the sets of numbers to which each apply. (a)√ 16 = 4 - N, W, Z, Q, R (b)-185 - Z, Q, R (c)√ 20 - I, R (d) -⅞ - Q, R __ (e) 0.45 - Q, R

Properties of Real Numbers PropertyAdditionMultiplication Commutativea + b = b + aa·b = b·a Associative (a+b)+c = a+(b+c) (a · b) · c = a · (b · c) Identitya+0 = a = 0+aa·1 = a = 1·a Inversea+(-a) =0= -a+aa·1 =1= 1·a a a Distributivea(b+c)=ab+ac and (b+c)a=ba+ca

Example 2

Name the property used in each equation.

Example 2 Name the property used in each equation. (a) (5 + 7) + 8 = 8 + (5 + 7)

Example 2 Name the property used in each equation. (a) (5 + 7) + 8 = 8 + (5 + 7) Commutative Addition

Example 2 Name the property used in each equation. (a) (5 + 7) + 8 = 8 + (5 + 7) Commutative Addition (b) 3(4x) = (3·4)x

Example 2 Name the property used in each equation. (a) (5 + 7) + 8 = 8 + (5 + 7) Commutative Addition (b) 3(4x) = (3·4)x Associative Multiplication

Example 3 What is the additive and multiplicative inverse for -1¾?

Example 3 What is the additive and multiplicative inverse for -1¾? Additive: -1¾

Example 3 What is the additive and multiplicative inverse for -1¾? Additive: -1¾ + = 0

Example 3 What is the additive and multiplicative inverse for -1¾? Additive: -1¾ + 1¾ = 0

Example 3 What is the additive and multiplicative inverse for -1¾? Additive: -1¾ + 1¾ = 0 Multiplicative: -1¾

Example 3 What is the additive and multiplicative inverse for -1¾? Additive: -1¾ + 1¾ = 0 Multiplicative: -1¾ · = 1

Example 3 What is the additive and multiplicative inverse for -1¾? Additive: -1¾ + 1¾ = 0 Multiplicative: (-1¾)(- 4 / 7 ) = 1

Example 4

Simplify 2(5m+n)+3(2m–4n).

Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n)

Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n)

Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)

Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+

Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+

Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)

Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)+

Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)+

Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)+3(2m)

Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)+3(2m)-

Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)+3(2m)-

Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)+3(2m)-3(4n)

Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)+3(2m)-3(4n) 10m

Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)+3(2m)-3(4n) 10m +

Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)+3(2m)-3(4n) 10m + 2n

Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)+3(2m)-3(4n) 10m + 2n +

Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)+3(2m)-3(4n) 10m + 2n + 6m

Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)+3(2m)-3(4n) 10m + 2n + 6m –

Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)+3(2m)-3(4n) 10m + 2n + 6m – 12n

Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)+3(2m)-3(4n) 10m + 2n + 6m – 12n

Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)+3(2m)-3(4n) 10m + 2n + 6m – 12n

Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)+3(2m)-3(4n) 10m + 2n + 6m – 12n 10m + 6m + 2n – 12n

Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)+3(2m)-3(4n) 10m + 2n + 6m – 12n 10m + 6m + 2n – 12n 16m

Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)+3(2m)-3(4n) 10m + 2n + 6m – 12n 10m + 6m + 2n – 12n 16m – 10n

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