
1.3E: Exercises


Practice Makes Perfect

Use Variables and Algebraic Symbols

In the following exercises, translate from algebra to English.

Exercise $$\PageIndex{55}$$

$$16−9$$

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

Exercise $$\PageIndex{56}$$

$$3\cdot 9$$

Exercise $$\PageIndex{57}$$

$$28\div 4$$

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

Exercise $$\PageIndex{58}$$

$$x+11$$

Exercise $$\PageIndex{59}$$

$$(2)(7)$$

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

Exercise $$\PageIndex{60}$$

$$(4)(8)$$

Exercise $$\PageIndex{61}$$

$$14<21$$

fourteen is less than twenty-one

Exercise $$\PageIndex{62}$$

$$17<35$$

Exercise $$\PageIndex{63}$$

$$36\geq 19$$

thirty-six is greater than or equal to nineteen

Exercise $$\PageIndex{64}$$

$$6n=36$$

Exercise $$\PageIndex{65}$$

$$y−1>6$$

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

Exercise $$\PageIndex{66}$$

$$y−4>8$$

Exercise $$\PageIndex{67}$$

$$2\leq 18\div 6$$

$$2$$ 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 $$\PageIndex{69}$$

$$9\cdot 6=54$$

equation

Exercise $$\PageIndex{70}$$

$$7\cdot 9=63$$

Exercise $$\PageIndex{71}$$

$$5\cdot 4+3$$

expression

Exercise $$\PageIndex{72}$$

$$x+7$$

Exercise $$\PageIndex{73}$$

$$x + 9$$

expression

Exercise $$\PageIndex{74}$$

$$y−5=25$$

Simplify Expressions Using the Order of Operations

In the following exercises, simplify each expression.

Exercise $$\PageIndex{75}$$

$$5^{3}$$

$$125$$

Exercise $$\PageIndex{76}$$

$$8^{3}$$

Exercise $$\PageIndex{77}$$

$$2^{8}$$

$$256$$

Exercise $$\PageIndex{78}$$

$$10^{5}$$

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

Exercise $$\PageIndex{79}$$

1. $$3+8\cdot 5$$
2. $$(3+8)\cdot 5$$
1. $$43$$
2. $$55$$

Exercise $$\PageIndex{80}$$

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

Exercise $$\PageIndex{81}$$

$$2^{3}−12\div (9−5)$$

$$5$$

Exercise $$\PageIndex{82}$$

$$3^{2}−18\div(11−5)$$

Exercise $$\PageIndex{83}$$

$$3\cdot 8+5\cdot 2$$

$$34$$

Exercise $$\PageIndex{84}$$

$$4\cdot 7+3\cdot 5$$

Exercise $$\PageIndex{85}$$

$$2+8(6+1)$$

$$58$$

Exercise $$\PageIndex{86}$$

$$4+6(3+6)$$

Exercise $$\PageIndex{87}$$

$$4\cdot 12/8$$

$$6$$

Exercise $$\PageIndex{88}$$

$$2\cdot 36/6$$

Exercise $$\PageIndex{89}$$

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

$$4$$

Exercise $$\PageIndex{90}$$

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

Exercise $$\PageIndex{91}$$

$$20\div4+6\cdot5$$

$$35$$

Exercise $$\PageIndex{92}$$

$$33\div3+8\cdot2$$

Exercise $$\PageIndex{93}$$

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

$$58$$

Exercise $$\PageIndex{94}$$

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

Exercise $$\PageIndex{95}$$

$$3(1+9\cdot6)−4^{2}$$

$$149$$

Exercise $$\PageIndex{96}$$

$$5(2+8\cdot4)−7^{2}$$

Exercise $$\PageIndex{97}$$

$$2[1+3(10−2)]$$

$$50$$

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 $$x=2$$

$$22$$

Exercise $$\PageIndex{100}$$

$$8x−6$$ when $$x=7$$

Exercise $$\PageIndex{101}$$

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

$$144$$

Exercise $$\PageIndex{102}$$

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

Exercise $$\PageIndex{103}$$

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

$$32$$

Exercise $$\PageIndex{104}$$

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

Exercise $$\PageIndex{105}$$

$$x^{2}+3x−7$$ when $$x = 4$$

$$21$$

Exercise $$\PageIndex{106}$$

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

$$9$$

Exercise $$\PageIndex{107}$$

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

Exercise $$\PageIndex{108}$$

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

$$73$$

Exercise $$\PageIndex{109}$$

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

Exercise $$\PageIndex{110}$$

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

$$54$$

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 $$\PageIndex{112}$$

$$8a$$

$$8$$

Exercise $$\PageIndex{113}$$

$$13m$$

Exercise $$\PageIndex{114}$$

$$5r^{2}$$

$$5$$

Exercise $$\PageIndex{115}$$

$$6x^{3}$$

In the following exercises, identify the like terms.

Exercise $$\PageIndex{116}$$

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

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

Exercise $$\PageIndex{117}$$

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

Exercise $$\PageIndex{118}$$

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

$$16$$ 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 $$\PageIndex{120}$$

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

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

Exercise $$\PageIndex{121}$$

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

Exercise $$\PageIndex{122}$$

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

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

Exercise $$\PageIndex{123}$$

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

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

Exercise $$\PageIndex{124}$$

$$10x+3x$$

$$13x$$

Exercise $$\PageIndex{125}$$

$$15x+4x$$

Exercise $$\PageIndex{126}$$

$$4c + 2c + c$$

$$7c$$

Exercise $$\PageIndex{127}$$

$$6y + 4y + y$$

Exercise $$\PageIndex{128}$$

$$7u + 2 + 3u + 1$$

$$10u + 3$$

Exercise $$\PageIndex{129}$$

$$8d + 6 + 2d + 5$$

Exercise $$\PageIndex{130}$$

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

$$22a + 1$$

Exercise $$\PageIndex{131}$$

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

Exercise $$\PageIndex{132}$$

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

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

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 $$9$$

$$14−9$$

Exercise $$\PageIndex{135}$$

the difference of $$19$$ and $$8$$

Exercise $$\PageIndex{136}$$

the product of $$9$$ and $$7$$

$$9\cdot 7$$

Exercise $$\PageIndex{137}$$

the product of $$8$$ and $$7$$

Exercise $$\PageIndex{138}$$

the quotient of $$36$$ and $$9$$

$$36\div 9$$

Exercise $$\PageIndex{139}$$

the quotient of $$42$$ and $$7$$

Exercise $$\PageIndex{140}$$

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

$$8x+3x$$

Exercise $$\PageIndex{141}$$

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

Exercise $$\PageIndex{142}$$

the quotient of $$y$$ and $$3$$

$$\frac{y}{3}$$

Exercise $$\PageIndex{143}$$

the quotient of $$y$$ and $$8$$

Exercise $$\PageIndex{144}$$

eight times the difference of $$y$$ and nine

$$8(y−9)$$

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.

$$c+3$$

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.

$$2n - 7$$

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?
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.

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 $$8a^{2} + 4a + 9 - a^{2} - 1$$

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 $$y$$.”