Algebra for College Students

Margaret L. Lial, John Hornsby, Terry McGinnis

Chapter 1

Linear Equations, Inequalities, and Applications - all with Video Answers

Educators

Section 1

Linear Equations in One Variable

Problem 1

Complete each statement. The following key terms may be used once, more than once, or not at all.
$$\begin{array}{cc}\text{linear equation}&\text{solution}&\text{algebraic expression}&\text{contradiction}& \text{all real numbers}\\ \text{solution set}&\text{identity}&\text{conditional equation}&\text{first-degree equation }&\text{empty set}\varnothing
\end{array}$$
A collection of numbers, variables, operation symbols, and grouping symbols, such as $2(8 x-15),$ is a $(n)$ __________. While an equation (does/does not) include an equality symbol, there (is / is not) an equality symbol in an algebraic expression.

Maryam Shahid

Maryam Shahid

Numerade Educator

Problem 2

Complete each statement. The following key terms may be used once, more than once, or not at all.
$$\begin{array}{cc}\text{linear equation}&\text{solution}&\text{algebraic expression}&\text{contradiction}& \text{all real numbers}\\ \text{solution set}&\text{identity}&\text{conditional equation}&\text{first-degree equation }&\text{empty set}\varnothing
\end{array}$$
$\mathrm{A}(\mathrm{n})$ __________ in one variable (here $x$ ) can be written in the form $a x+b(=/>/<)$ 0, with $a \neq 0$. Another name for a linear equation is a(n) _________, because the greatest power on the variable is (one/two/three).

Avinash Vishwakarma

Avinash Vishwakarma

Numerade Educator

Problem 3

Complete each statement. The following key terms may be used once, more than once, or not at all.
$$\begin{array}{cc}\text{linear equation}&\text{solution}&\text{algebraic expression}&\text{contradiction}& \text{all real numbers}\\ \text{solution set}&\text{identity}&\text{conditional equation}&\text{first-degree equation }&\text{empty set}\varnothing
\end{array}$$
If we let $x=2$ in the linear equation $2 x+5=9,$ a (true/false) statement results. The number 2 is a(n) __________ of the equation, and \{2\} is the ____________.

Maryam Shahid

Numerade Educator

Problem 4

Complete each statement. The following key terms may be used once, more than once, or not at all.
$$\begin{array}{cc}\text{linear equation}&\text{solution}&\text{algebraic expression}&\text{contradiction}& \text{all real numbers}\\ \text{solution set}&\text{identity}&\text{conditional equation}&\text{first-degree equation }&\text{empty set}\varnothing
\end{array}$$
A linear equation with one solution in its ____________, such as $2 x+5=9,$ is $\mathrm{a}(\mathrm{n})$ ___________.

Maryam Shahid

Numerade Educator

Problem 5

A linear equation with an infinite number of solutions is a(n)___________. Its solution set is {_____________}.

Maryam Shahid

Numerade Educator

Problem 6

A linear equation with no solution is a(n) ____________. Its solution set is the ___________.

Maryam Shahid

Numerade Educator

Problem 7

Which equations are linear equations in $x ?$
A. $3 x+x-2=0$
B. $12=x^{2}$
C. $9 x-4=9$
D. $3 x+2 y=6$

Anas Venkitta

Numerade Educator

Problem 8

Which of the equations in Exercise 7 are nonlinear equations in $x ?$ Explain why.

Rupsa Sarkar

Numerade Educator

Problem 9

Determine whether each of the following is an expression or an equation.
$-3 x+2-4=x$

Anas Venkitta

Numerade Educator

Problem 10

Determine whether each of the following is an expression or an equation.
$-3 x+2-4-x=4$

Anas Venkitta

Numerade Educator

Problem 11

Determine whether each of the following is an expression or an equation.
$4(x+3)-2(x+1)-10$

Anas Venkitta

Numerade Educator

Problem 12

Determine whether each of the following is an expression or an equation.
$4(x+3)-2(x+1)+10$

Anas Venkitta

Numerade Educator

Problem 13

Determine whether each of the following is an expression or an equation.
$-10 x+12-4 x=-3$

Anas Venkitta

Numerade Educator

Problem 14

Determine whether each of the following is an expression or an equation.
$-10 x+12-4 x+3=0$

Anas Venkitta

Numerade Educator

Problem 15

This incorrect solution contains a common error.
$\begin{aligned} 8 x-2(2 x-3) &=3 x+7 \\ 8 x-4 x-6 &=3 x+7 \text { Distributive property }\\
&4 x-6=3 x+7 \quad \text {Combine like terms}\\
&x=13 \quad \text {Subtract 3 x . Add 6} .
\end{aligned}$
WHAT WENT WRONG? Give the correct solution.

Maryam Shahid

Numerade Educator

Problem 16

This incorrect solution contains a common error
$\begin{aligned} 12-2(3 x+1) &=11 \\ 10(3 x+1) &=11 \quad \text { Subtract } 12-2 . \end{aligned}$
$\begin{aligned} 30 x &=1 & & \text { Subtract } 10 \\ x &=\frac{1}{30} & & \text { Divide by } 30 . \end{aligned}$

Maryam Shahid

Numerade Educator

Problem 17

Concept Check Suppose we solve a linear equation and obtain, as our final result, an equation in Column I. Match each result with the solution set in Column II for the original equation.
$\mathbf{I}$
(a) $7=7$
(b) $x=0$
(c) $7=0$
$\mathbf{II}$
A. $\{0\}$
B. $\{$ all real numbers $\}$
C. $\varnothing$

Maryam Shahid

Numerade Educator

Problem 18

Which one of the following linear equations does not have solution set $\{$ all real numbers $\} ?$
A. $4 x=5 x-x$
B. $3(x+4)=3 x+12$
C. $4 x=3 x$
D. $\frac{3}{4} x=0.75 x$
E. $4(x-2)=2(2 x-4)$
F. $2 x+18 x=20 x$

Deepak Kumar

Numerade Educator

Problem 19

Solve each equation, and check the solution. If applicable, tell whether the equation is an identity or a contradiction.
$7 x+8=1$

Anas Venkitta

Numerade Educator

Problem 20

Solve each equation, and check the solution. If applicable, tell whether the equation is an identity or a contradiction.
$5 x-4=21$

Anas Venkitta

Numerade Educator

Problem 21

Solve each equation, and check the solution. If applicable, tell whether the equation is an identity or a contradiction.
$5 x+2=3 x-6$

Anas Venkitta

Numerade Educator

Problem 22

Solve each equation, and check the solution. If applicable, tell whether the equation is an identity or a contradiction.
$9 x+1=7 x-9$

Anas Venkitta

Numerade Educator

Problem 23

Solve each equation, and check the solution. If applicable, tell whether the equation is an identity or a contradiction.
$7 x-5 x+15=x+8$

Anas Venkitta

Numerade Educator

Problem 24

Solve each equation, and check the solution. If applicable, tell whether the equation is an identity or a contradiction.
$2 x+4-x=4 x-5$

Anas Venkitta

Numerade Educator

Problem 25

Solve each equation, and check the solution. If applicable, tell whether the equation is an identity or a contradiction.
$12 w+15 w-9+5=-3 w+5-9$

Anas Venkitta

Numerade Educator

Problem 26

Solve each equation, and check the solution. If applicable, tell whether the equation is an identity or a contradiction.
$-4 x+5 x-8+4=6 x-4$

Anas Venkitta

Numerade Educator

Problem 27

Solve each equation, and check the solution. If applicable, tell whether the equation is an identity or a contradiction.
$3(2 t-4)=20-2 t$

Avinash Vishwakarma

Avinash Vishwakarma

Numerade Educator

Problem 28

Solve each equation, and check the solution. If applicable, tell whether the equation is an identity or a contradiction.
$2(3-2 x)=x-4$

Avinash Vishwakarma

Avinash Vishwakarma

Numerade Educator

Problem 29

Solve each equation, and check the solution. If applicable, tell whether the equation is an identity or a contradiction.
$-5(x+1)+3 x+2=6 x+4$

Avinash Vishwakarma

Avinash Vishwakarma

Numerade Educator

Problem 30

Solve each equation, and check the solution. If applicable, tell whether the equation is an identity or a contradiction.
$5(x+3)+4 x-5=4-2 x$

Deepak Kumar

Numerade Educator

Problem 31

Solve each equation, and check the solution. If applicable, tell whether the equation is an identity or a contradiction.
$-2 x+5 x-9=3(x-4)-5$

Deepak Kumar

Numerade Educator

Problem 32

Solve each equation, and check the solution. If applicable, tell whether the equation is an identity or a contradiction.
$-6 x+2 x-11=-2(2 x-3)+4$

Deepak Kumar

Numerade Educator

Problem 33

Solve each equation, and check the solution. If applicable, tell whether the equation is an identity or a contradiction.
$-2(x+3)=-6(x+7)$

Avinash Vishwakarma

Avinash Vishwakarma

Numerade Educator

Problem 34

Solve each equation, and check the solution. If applicable, tell whether the equation is an identity or a contradiction.
$-4(x-9)=-8(x+3)$

Deepak Kumar

Numerade Educator

Problem 35

Solve each equation, and check the solution. If applicable, tell whether the equation is an identity or a contradiction.
$3(2 x+1)-2(x-2)=5$

Deepak Kumar

Numerade Educator

Problem 36

Solve each equation, and check the solution. If applicable, tell whether the equation is an identity or a contradiction.
$4(x-2)+2(x+3)=6$

Deepak Kumar

Numerade Educator

Problem 37

Solve each equation, and check the solution. If applicable, tell whether the equation is an identity or a contradiction.
$2 x+3(x-4)=2(x-3)$

Deepak Kumar

Numerade Educator

Problem 38

Solve each equation, and check the solution. If applicable, tell whether the equation is an identity or a contradiction.
$6 x-3(5 x+2)=4(1-x)$

Deepak Kumar

Numerade Educator

Problem 39

Solve each equation, and check the solution. If applicable, tell whether the equation is an identity or a contradiction.
$6 x-4(3-2 x)=5(x-4)-10$

Deepak Kumar

Numerade Educator

Problem 40

Solve each equation, and check the solution. If applicable, tell whether the equation is an identity or a contradiction.
$-2 x-3(4-2 x)=2(x-3)+2$

Deepak Kumar

Numerade Educator

Problem 41

Solve each equation, and check the solution. If applicable, tell whether the equation is an identity or a contradiction.
$-2(x+3)-x-4=-3(x+4)+2$

Deepak Kumar

Numerade Educator

Problem 42

Solve each equation, and check the solution. If applicable, tell whether the equation is an identity or a contradiction.
$4(2 x+7)=2 x+25+3(2 x+1)$

Deepak Kumar

Numerade Educator

Problem 43

Solve each equation, and check the solution. If applicable, tell whether the equation is an identity or a contradiction.
$2[x-(2 x+4)+3]=2(x+1)$

Deepak Kumar

Numerade Educator

Problem 44

Solve each equation, and check the solution. If applicable, tell whether the equation is an identity or a contradiction.
$4[2 x-(3-x)+5]=-(2+7 x)$

Deepak Kumar

Numerade Educator

Problem 45

Solve each equation, and check the solution. If applicable, tell whether the equation is an identity or a contradiction.
$-[2 x-(5 x+2)]=2+(2 x+7)$

Deepak Kumar

Numerade Educator

Problem 46

Solve each equation, and check the solution. If applicable, tell whether the equation is an identity or a contradiction.
$-[6 x-(4 x+8)]=9+(6 x+3)$

Deepak Kumar

Numerade Educator

Problem 47

Solve each equation, and check the solution. If applicable, tell whether the equation is an identity or a contradiction.
$-3 x+6-5(x-1)=-5 x-(2 x-4)+5$

Deepak Kumar

Numerade Educator

Problem 48

Solve each equation, and check the solution. If applicable, tell whether the equation is an identity or a contradiction.
$4(x+2)-8 x-5=-3 x+9-2(x+6)$

Deepak Kumar

Numerade Educator

Problem 49

Solve each equation, and check the solution. If applicable, tell whether the equation is an identity or a contradiction.
$7[2-(3+4 x)]-2 x=-9+2(1-15 x)$

Deepak Kumar

Numerade Educator

Problem 50

Solve each equation, and check the solution. If applicable, tell whether the equation is an identity or a contradiction.
$4[6-(1+2 x)]+10 x=2(10-3 x)+8 x$

Deepak Kumar

Numerade Educator

Problem 51

To solve the following linear equation, we multiply each side by the least common denominator of all the fractions in the equation. What is this $L C D ?$
$$
\frac{3}{4} x-\frac{1}{3} x=\frac{5}{6} x-5
$$

Deepak Kumar

Numerade Educator

Problem 52

Suppose that in solving the following equation, we multiply each side by $12,$ rather than the least common denominator, $6 .$ Would we obtain the correct solution? Explain.
$$
\frac{1}{3} x+\frac{1}{2} x=\frac{1}{6} x
$$

Deepak Kumar

Numerade Educator

Problem 53

To solve a linear equation with decimals, we usually begin by multiplying by a power of 10 so that all coefficients are integers. What is the least power of 10 that will accomplish this goal in each equation?
(a) $0.05 x+0.12(x+5000)=940$
(b) $0.006(x+2)=0.007 x+0.009$

Deepak Kumar

Numerade Educator

Problem 54

The expression $0.06(10-x)(100)$ is equivalent to which one of the following?
A. $0.06-0.06 x$
B. $60-6 x$
C. $6-6 x$
D. $6-0.06 x$

Avinash Vishwakarma

Avinash Vishwakarma

Numerade Educator

Problem 55

Solve each equation, and check the solution.
$-\frac{5}{9} x=2$

Avinash Vishwakarma

Avinash Vishwakarma

Numerade Educator

Problem 56

Solve each equation, and check the solution.
$\frac{3}{11} x=-5$

Avinash Vishwakarma

Avinash Vishwakarma

Numerade Educator

Problem 57

Solve each equation, and check the solution.
$\frac{6}{5} x=-1$

Avinash Vishwakarma

Avinash Vishwakarma

Numerade Educator

Problem 58

Solve each equation, and check the solution.
$-\frac{7}{8} x=6$

Avinash Vishwakarma

Avinash Vishwakarma

Numerade Educator

Problem 59

Solve each equation, and check the solution.
$\frac{x}{2}+\frac{x}{3}=5$

Avinash Vishwakarma

Avinash Vishwakarma

Numerade Educator

Problem 60

Solve each equation, and check the solution.
$\frac{x}{5}-\frac{x}{4}=1$

Avinash Vishwakarma

Avinash Vishwakarma

Numerade Educator

Problem 61

Solve each equation, and check the solution.
$\frac{3 x}{4}+\frac{5 x}{2}=13$

Avinash Vishwakarma

Avinash Vishwakarma

Numerade Educator

Problem 62

Solve each equation, and check the solution.
$\frac{8 x}{3}-\frac{x}{2}=-13$

Avinash Vishwakarma

Avinash Vishwakarma

Numerade Educator

Problem 63

Solve each equation, and check the solution.
$\frac{x-10}{5}+\frac{2}{5}=-\frac{x}{3}$

Avinash Vishwakarma

Avinash Vishwakarma

Numerade Educator

Problem 64

Solve each equation, and check the solution.
$\frac{5-x}{6}+\frac{5}{6}=\frac{x}{54}$

Avinash Vishwakarma

Avinash Vishwakarma

Numerade Educator

Problem 65

Solve each equation, and check the solution.
$\frac{3 x-1}{4}+\frac{x+3}{6}=3$

Avinash Vishwakarma

Avinash Vishwakarma

Numerade Educator

Problem 66

Solve each equation, and check the solution.
$\frac{3 x+2}{7}-\frac{x+4}{5}=2$

Avinash Vishwakarma

Avinash Vishwakarma

Numerade Educator

Problem 67

Solve each equation, and check the solution.
$\frac{4 x+1}{3}=\frac{x+5}{6}+\frac{x-3}{6}$

Avinash Vishwakarma

Avinash Vishwakarma

Numerade Educator

Problem 68

Solve each equation, and check the solution.
$\frac{2 x+5}{5}=\frac{3 x+1}{2}+\frac{-x+8}{16}$

Deepak Kumar

Numerade Educator

Problem 69

Solve each equation, and check the solution.
$0.04 x+0.06+0.03 x=0.03 x+1.46$

Avinash Vishwakarma

Avinash Vishwakarma

Numerade Educator

Problem 70

Solve each equation, and check the solution.
$0.05 x+0.08+0.06 x=0.07 x+0.68$

Avinash Vishwakarma

Avinash Vishwakarma

Numerade Educator

Problem 71

Solve each equation, and check the solution.
$0.006 x-0.02 x+0.03=0.008 x+0.25$

Deepak Kumar

Numerade Educator

Problem 72

Solve each equation, and check the solution.
$0.05 x-0.1 x+0.6=0.04 x+2.22$

Deepak Kumar

Numerade Educator

Problem 73

Solve each equation, and check the solution.
$0.05 x+0.12(x+5000)=940$

Avinash Vishwakarma

Avinash Vishwakarma

Numerade Educator

Problem 74

Solve each equation, and check the solution.
$0.09 x+0.13(x+300)=61$

Avinash Vishwakarma

Avinash Vishwakarma

Numerade Educator

Problem 75

Solve each equation, and check the solution.
$0.02(50)+0.08 x=0.04(50+x)$

Deepak Kumar

Numerade Educator

Problem 76

Solve each equation, and check the solution.
$0.04(90)+0.12 x=0.06(460+x)$

Deepak Kumar

Numerade Educator

Problem 77

Solve each equation, and check the solution.
$0.05 x+0.10(200-x)=0.45 x$

Deepak Kumar

Numerade Educator

Problem 78

Solve each equation, and check the solution.
$0.08 x+0.12(260-x)=0.48 x$

Deepak Kumar

Numerade Educator

Problem 79

Solve each equation, and check the solution.
$0.006(x+2)=0.007 x+0.009$

Deepak Kumar

Numerade Educator

Problem 80

Solve each equation, and check the solution.
$0.006(50-x)=0.272-0.004 x$

Deepak Kumar

Numerade Educator