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Mathematics LibreTexts

1.3E: Exercises

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    30455
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    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.

    A table is shown that is composed of four columns and six rows. The header row reads, from left to right, “I can …”, “Confidently”, “With some help” and “No – I don’t get it!”. The phrases in the first column read “use variables and algebraic symbols.”, “simplify expressions using the order of operations.”, “evaluate an expression.”, “identify and combine like terms.”, and “translate English phrases to algebraic expressions.”

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