1.3E: Exercises

Practice Makes Perfect

Use Variables and Algebraic Symbols

In the following exercises, translate from algebra to English.

Exercise $\PageIndex{55}$

$16−9$

Answer

$16$ minus $9$, the difference of sixteen and nine

Exercise $\PageIndex{56}$

$3\cdot 9$

Exercise $\PageIndex{57}$

$28\div 4$

Answer

$28$ divided by $4$, the quotient of twenty-eight and four

Exercise $\PageIndex{58}$

$x+11$

Exercise $\PageIndex{59}$

$(2)(7)$

Answer

$2$ times $7$, the product of two and seven

Exercise $\PageIndex{60}$

$(4)(8)$

Exercise $\PageIndex{61}$

$14<21$

Answer

fourteen is less than twenty-one

Exercise $\PageIndex{62}$

$17<35$

Exercise $\PageIndex{63}$

$36\geq 19$

Answer

thirty-six is greater than or equal to nineteen

Exercise $\PageIndex{64}$

$6n=36$

Exercise $\PageIndex{65}$

$y−1>6$

Answer

$y$ minus $1$ is greater than $6$, the difference of $y$ and one is greater than six

Exercise $\PageIndex{66}$

$y−4>8$

Exercise $\PageIndex{67}$

$2\leq 18\div 6$

Answer

$2$ is less than or equal to $18$ divided by $6$; $2$ is less than or equal to the quotient of eighteen and six

Exercise $\PageIndex{68}$

$a\neq 1\cdot12$

In the following exercises, determine if each is an expression or an equation.

Exercise $\PageIndex{69}$

$9\cdot 6=54$

Answer

equation

Exercise $\PageIndex{70}$

$7\cdot 9=63$

Exercise $\PageIndex{71}$

$5\cdot 4+3$

Answer

expression

Exercise $\PageIndex{72}$

$x+7$

Exercise $\PageIndex{73}$

$x + 9$

Answer

expression

Exercise $\PageIndex{74}$

$y−5=25$

Simplify Expressions Using the Order of Operations

In the following exercises, simplify each expression.

Exercise $\PageIndex{75}$

$5^{3}$

Answer

$125$

Exercise $\PageIndex{76}$

$8^{3}$

Exercise $\PageIndex{77}$

$2^{8}$

Answer

$256$

Exercise $\PageIndex{78}$

$10^{5}$

In the following exercises, simplify using the order of operations.

Exercise $\PageIndex{79}$

1. $3+8\cdot 5$
2. $(3+8)\cdot 5$

Answer

1. $43$
2. $55$

Exercise $\PageIndex{80}$

1. $2+6\cdot 3$
2. $(2+6)\cdot 3$

Exercise $\PageIndex{81}$

$2^{3}−12\div (9−5)$

Answer

$5$

Exercise $\PageIndex{82}$

$3^{2}−18\div(11−5)$

Exercise $\PageIndex{83}$

$3\cdot 8+5\cdot 2$

Answer

$34$

Exercise $\PageIndex{84}$

$4\cdot 7+3\cdot 5$

Exercise $\PageIndex{85}$

$2+8(6+1)$

Answer

$58$

Exercise $\PageIndex{86}$

$4+6(3+6)$

Exercise $\PageIndex{87}$

$4\cdot 12/8$

Answer

$6$

Exercise $\PageIndex{88}$

$2\cdot 36/6$

Exercise $\PageIndex{89}$

$(6+10)\div(2+2)$

Answer

$4$

Exercise $\PageIndex{90}$

$(9+12)\div(3+4)$

Exercise $\PageIndex{91}$

$20\div4+6\cdot5$

Answer

$35$

Exercise $\PageIndex{92}$

$33\div3+8\cdot2$

Exercise $\PageIndex{93}$

$3^{2}+7^{2}$

Answer

$58$

Exercise $\PageIndex{94}$

$(3+7)^{2}$

Exercise $\PageIndex{95}$

$3(1+9\cdot6)−4^{2}$

Answer

$149$

Exercise $\PageIndex{96}$

$5(2+8\cdot4)−7^{2}$

Exercise $\PageIndex{97}$

$2[1+3(10−2)]$

Answer

$50$

Exercise $\PageIndex{98}$

$5[2+4(3−2)]$

Evaluate an Expression

In the following exercises, evaluate the following expressions.

Exercise $\PageIndex{99}$

$7x+8$ when $x=2$

Answer

$22$

Exercise $\PageIndex{100}$

$8x−6$ when $x=7$

Exercise $\PageIndex{101}$

$x^{2}$ when $x = 12$

Answer

$144$

Exercise $\PageIndex{102}$

$x^{3}$ when $x = 5$

Exercise $\PageIndex{103}$

$x^{5}$ when $x = 2$

Answer

$32$

Exercise $\PageIndex{104}$

$4^{x}$ when $x = 2$

Exercise $\PageIndex{105}$

$x^{2}+3x−7$ when $x = 4$

Answer

$21$

Exercise $\PageIndex{106}$

$6x + 3y - 9$ when $x = 10, y = 7$

Answer

$9$

Exercise $\PageIndex{107}$

$(x + y)^{2}$ when $x = 6, y = 9$

Exercise $\PageIndex{108}$

$a^{2} + b^{2}$ when $a = 3, b = 8$

Answer

$73$

Exercise $\PageIndex{109}$

$r^{2} - s^{2}$ when $r = 12, s = 5$

Exercise $\PageIndex{110}$

$2l + 2w$ when $l = 15, w = 12$

Answer

$54$

Exercise $\PageIndex{111}$

$2l + 2w$ when $l = 18, w = 14$

Simplify Expressions by Combining Like Terms

In the following exercises, identify the coefficient of each term.

Exercise $\PageIndex{112}$

$8a$

Answer

$8$

Exercise $\PageIndex{113}$

$13m$

Exercise $\PageIndex{114}$

$5r^{2}$

Answer

$5$

Exercise $\PageIndex{115}$

$6x^{3}$

In the following exercises, identify the like terms.

Exercise $\PageIndex{116}$

$x^{3}, 8x, 14, 8y, 5, 8x^{3}$

Answer

$x^{3}$ and $8x^{3}$, $14$ and $5$

Exercise $\PageIndex{117}$

$6z, 3w^{2}, 1, 6z^{2}, 4z, w^{2}$

Exercise $\PageIndex{118}$

$9a, a^{2}, 16, 16b^{2}, 4, 9b^{2}$

Answer

$16$ and $4$, $16b^{2}$ and $9b^{2}$

Exercise $\PageIndex{119}$

$3, 25r^{2}, 10s, 10r, 4r^{2}, 3s$

In the following exercises, identify the terms in each expression.

Exercise $\PageIndex{120}$

$15x^{2} + 6x + 2$

Answer

$15x^{2}, 6x, 2$

Exercise $\PageIndex{121}$

$11x^{2} + 8x + 5$

Exercise $\PageIndex{122}$

$10y^{3} + y + 2$

Answer

$10y^{3}, y, 2$

Exercise $\PageIndex{123}$

$9y^{3} + y + 5$

In the following exercises, simplify the following expressions by combining like terms.

Exercise $\PageIndex{124}$

$10x+3x$

Answer

$13x$

Exercise $\PageIndex{125}$

$15x+4x$

Exercise $\PageIndex{126}$

$4c + 2c + c$

Answer

$7c$

Exercise $\PageIndex{127}$

$6y + 4y + y$

Exercise $\PageIndex{128}$

$7u + 2 + 3u + 1$

Answer

$10u + 3$

Exercise $\PageIndex{129}$

$8d + 6 + 2d + 5$

Exercise $\PageIndex{130}$

$10a + 7 + 5a - 2 + 7a - 4$

Answer

$22a + 1$

Exercise $\PageIndex{131}$

$7c + 4 + 6c - 3 + 9c - 1$

Exercise $\PageIndex{132}$

$3x^{2} + 12x + 11 + 14x^{2} + 8x + 5$

Answer

$17x^{2} + 20x + 16$

Exercise $\PageIndex{133}$

$5b^{2} + 9b + 10 + 2b^{2} + 3b - 4$

Translate an English Phrase to an Algebraic Expression

In the following exercises, translate the phrases into algebraic expressions.

Exercise $\PageIndex{134}$

the difference of $14$ and $9$

Answer

$14−9$

Exercise $\PageIndex{135}$

the difference of $19$ and $8$

Exercise $\PageIndex{136}$

the product of $9$ and $7$

Answer

$9\cdot 7$

Exercise $\PageIndex{137}$

the product of $8$ and $7$

Exercise $\PageIndex{138}$

the quotient of $36$ and $9$

Answer

$36\div 9$

Exercise $\PageIndex{139}$

the quotient of $42$ and $7$

Exercise $\PageIndex{140}$

the sum of $8x$ and $3x$

Answer

$8x+3x$

Exercise $\PageIndex{141}$

the sum of $13x$ and $3x$

Exercise $\PageIndex{142}$

the quotient of $y$ and $3$

Answer

$\frac{y}{3}$

Exercise $\PageIndex{143}$

the quotient of $y$ and $8$

Exercise $\PageIndex{144}$

eight times the difference of $y$ and nine

Answer

$8(y−9)$

Exercise $\PageIndex{145}$

seven times the difference of $y$ and one

Exercise $\PageIndex{146}$

Eric has rock and classical CDs in his car. The number of rock CDs is $3$ more than the number of classical CDs. Let $c$ represent the number of classical CDs. Write an expression for the number of rock CDs.

Answer

$c+3$

Exercise $\PageIndex{147}$

The number of girls in a second-grade class is $4$ less than the number of boys. Let $b$ represent the number of boys. Write an expression for the number of girls.

Exercise $\PageIndex{148}$

Greg has nickels and pennies in his pocket. The number of pennies is seven less than twice the number of nickels. Let $n$ represent the number of nickels. Write an expression for the number of pennies.

Answer

$2n - 7$

Exercise $\PageIndex{149}$

Jeannette has $5 and $10 bills in her wallet. The number of fives is three more than six times the number of tens. Let $t$ represent the number of tens. Write an expression for the number of fives.

Everyday Math

Exercise $\PageIndex{150}$

Car insurance Justin’s car insurance has a $750 deductible per incident. This means that he pays $750 and his insurance company will pay all costs beyond $750. If Justin files a claim for $2,100.

1. how much will he pay?
2. how much will his insurance company pay?

Answer

1. $750
2. $1,350

Exercise $\PageIndex{151}$

Home insurance Armando’s home insurance has a $2,500 deductible per incident. This means that he pays $2,500 and the insurance company will pay all costs beyond $2,500. If Armando files a claim for $19,400.

1. how much will he pay?
2. how much will the insurance company pay?

Writing Exercises

Exercise $\PageIndex{152}$

Explain the difference between an expression and an equation.

Answer

Answers may vary

Exercise $\PageIndex{153}$

Why is it important to use the order of operations to simplify an expression?

Exercise $\PageIndex{154}$

Explain how you identify the like terms in the expression $8a^{2} + 4a + 9 - a^{2} - 1$

Answer

Answers may vary

Exercise $\PageIndex{155}$

Explain the difference between the phrases “$4$ times the sum of $x$ and $y$” and “the sum of $4$ times $x$ and $y$.”

Self Check

ⓐ Use this checklist to evaluate your mastery of the objectives of this section.

ⓑ After reviewing this checklist, what will you do to become confident for all objectives?