Elementary and Intermediate Algebra 5th

Alan S. Tussy, R. David Gustafson

Chapter 4

Systems of Linear Equations and Inequalities - all with Video Answers

Educators

EB

RD

Section 5

Solving Systems of Linear Inequalities

Problem 1

Fill in the blanks.
$\left\{\begin{array}{l}{x+y>2} \\ {x+y<4}\end{array}\right.$ is a system of linear _____.

JO

Jorge O.

Numerade Educator

Problem 2

Fill in the blanks.
To graph the linear inequality $x+y>2,$ first graph the _____ $x+y=2 .$ Then pick the test _____ $(0,0)$ to determine which half-plane to shade.

JO

Jorge O.

Numerade Educator

Problem 3

Fill in the blanks.
To find the solutions of a system of two linear inequalities graphically, look for the _____, or overlap, of the two shaded regions.

JO

Jorge O.

Numerade Educator

Problem 4

Fill in the blanks.
The phrase should not surpass can be represented by the inequality symbol _____ and the phrase must be at least can be represented by the inequality symbol _____.

JO

Jorge O.

Numerade Educator

Problem 5

a. What is the equation of the boundary line of the graph of $3 x-y<5 ?$
b. Is the boundary a solid or dashed line?

JO

Jorge O.

Numerade Educator

Problem 6

a. What is the equation of the boundary line of the graph of $y \geq 4 x ?$
b. Is the boundary a solid or dashed line?
c. Why can't $(0,0)$ be used as a test point to determine what to shade?

JO

Jorge O.

Numerade Educator

Problem 7

Find the slope and the $y$ -intercept of the line whose equation is $y=4 x-3$.

JO

Jorge O.

Numerade Educator

Problem 8

Complete the table to find the $x$ - and $y$ -intercepts of the line whose equation is $8 x-3 y=-24$.
$$
\begin{array}{|c|c|}
\hline x & {y} \\
\hline 0 & {} \\
\hline & {0} \\
\hline
\end{array}
$$

JO

Jorge O.

Numerade Educator

Problem 9

The boundary of the graph of $2 x+y>4$ is shown.
a. Does the point $(0,0)$ make the inequality true?
b. Should the region above or below the boundary be shaded?
(GRAPH CANNOT COPY)

JO

Jorge O.

Numerade Educator

Problem 10

Linear inequality 1 is graphed in red below and linear inequality 2 is graphed in blue. Determine whether a true or false statement results when
a. The coordinates of point $A$ are substituted into inequality 1
b. The coordinates of point $A$ are substituted into inequality 2
c. The coordinates of point $B$ are substituted into inequality 1
d. The coordinates of point $B$ are substituted into inequality 2
e. The coordinates of point $C$ are substituted into inequality 1
f. The coordinates of point $C$ are substituted into inequality 2
(GRAPH CANNOT COPY)

JO

Jorge O.

Numerade Educator

Problem 11

The graph of a system of two linear inequalities is shown. Determine whether each point is a solution of the system.
a. $(4,-2)$
b. $(1,3)$
c. the origin
(GRAPH CANNOT COPY)

JO

Jorge O.

Numerade Educator

Problem 12

Use a check to determine whether each ordered pair is a solution of the system.
$$
\left\{\begin{array}{l}
{x+2 y \geq-1} \\
{x-y<2}
\end{array}\right.
$$
a. $(1,4)$
b. $(-2,0)$

JO

Jorge O.

Numerade Educator

Problem 13

Match each equation, inequality, or system with the graph of its solution.
a. $x+y=2$
b. $x+y \geq 2$
c. $\left\{\begin{array}{l}{x+y=2} \\ {x-y=2}\end{array}\right.$
d. $\left\{\begin{array}{l}{x+y \geq 2} \\ {x-y \leq 2}\end{array}\right.$

JO

Jorge O.

Numerade Educator

Problem 14

Match the system of inequalities with the correct graph.
a. $\left\{\begin{array}{l}{x \geq 2} \\ {y<1}\end{array}\right.$
b. $\left\{\begin{array}{l}{x>2} \\ {y \leq 1}\end{array}\right.$
c. $\left\{\begin{array}{l}{x \geq 2} \\ {y \geq 1}\end{array}\right.$
d. $\left\{\begin{array}{l}{x>2} \\ {y>-1}\end{array}\right.$

JO

Jorge O.

Numerade Educator

Problem 15

Graph the solutions of each system. See Example 1.
$$
\left\{\begin{array}{l}
{x+2 y \leq 3} \\
{2 x-y \geq 1}
\end{array}\right.
$$

JO

Jorge O.

Numerade Educator

Problem 16

Graph the solutions of each system. See Example 1.
$$
\left\{\begin{array}{l}
{2 x+y \geq 3} \\
{x-2 y \leq-1}
\end{array}\right.
$$

JO

Jorge O.

Numerade Educator

Problem 17

Graph the solutions of each system. See Example 1.
$$
\left\{\begin{array}{l}
{x+y<-1} \\
{x-y>-1}
\end{array}\right.
$$

JO

Jorge O.

Numerade Educator

Problem 18

Graph the solutions of each system. See Example 1.
$$
\left\{\begin{array}{l}
{x+y>2} \\
{x-y<-2}
\end{array}\right.
$$

JO

Jorge O.

Numerade Educator

Problem 19

Graph the solutions of each system. See Example 2.
$$
\left\{\begin{array}{l}
{y>2 x} \\
{x+2 y<6}
\end{array}\right.
$$

JO

Jorge O.

Numerade Educator

Problem 20

Graph the solutions of each system. See Example 2.
$$
\left\{\begin{array}{l}
{y \leq 2 x} \\
{x+y<4}
\end{array}\right.
$$

JO

Jorge O.

Numerade Educator

Problem 21

Graph the solutions of each system. See Example 2.
$$
\left\{\begin{array}{l}
{y \geq x} \\
{y \leq \frac{1}{3} x+1}
\end{array}\right.
$$

JO

Jorge O.

Numerade Educator

Problem 22

Graph the solutions of each system. See Example 2.
$$
\left\{\begin{array}{l}
{y>3 x} \\
{y \leq-x-1}
\end{array}\right.
$$

JO

Jorge O.

Numerade Educator

Problem 23

Graph the solutions of each system. See Example 3.
$$
\left\{\begin{array}{l}
{x \geq 2} \\
{y \leq 3}
\end{array}\right.
$$

JO

Jorge O.

Numerade Educator

Problem 24

Graph the solutions of each system. See Example 3.
$$
\left\{\begin{array}{l}
{x \geq-1} \\
{y>-2}
\end{array}\right.
$$

JO

Jorge O.

Numerade Educator

Problem 25

Graph the solutions of each system. See Example 3.
$$
\left\{\begin{array}{l}
{x>0} \\
{y>0}
\end{array}\right.
$$

JO

Jorge O.

Numerade Educator

Problem 26

Graph the solutions of each system. See Example 3.
$$
\left\{\begin{array}{l}
{x \leq 0} \\
{y<0}
\end{array}\right.
$$

JO

Jorge O.

Numerade Educator

Problem 27

Graph the solutions of each system. See Example 4.
$$
\left\{\begin{array}{l}
{x \geq 0} \\
{y \geq 0} \\
{x+y \leq 3}
\end{array}\right.
$$

JO

Jorge O.

Numerade Educator

Problem 28

Graph the solutions of each system. See Example 4.
$$
\left\{\begin{array}{l}
{x-y \leq 6} \\
{x+2 y \leq 6} \\
{x \geq 0}
\end{array}\right.
$$

JO

Jorge O.

Numerade Educator

Problem 29

Graph the solutions of each system. See Example 4.
$$
\left\{\begin{array}{l}
{x-y<4} \\
{y \leq 0} \\
{x \geq 0}
\end{array}\right.
$$

JO

Jorge O.

Numerade Educator

Problem 30

Graph the solutions of each system. See Example 4.
$$
\left\{\begin{array}{l}
{2 x+y \leq 2} \\
{y>x} \\
{x \geq 0}
\end{array}\right.
$$

Grace B.

Numerade Educator

Problem 31

Graph the solutions of each system.
$$
\left\{\begin{array}{l}
{2 x-3 y \leq 0} \\
{y \geq x-1}
\end{array}\right.
$$

Grace B.

Numerade Educator

Problem 32

Graph the solutions of each system.
$$
\left\{\begin{array}{l}
{y>2 x-4} \\
{y \geq-x-1}
\end{array}\right.
$$

Grace B.

Numerade Educator

Problem 33

Graph the solutions of each system.
$$
\left\{\begin{array}{l}
{x+y<2} \\
{x+y \leq 1}
\end{array}\right.
$$

Grace B.

Numerade Educator

Problem 34

Graph the solutions of each system.
$$
\left\{\begin{array}{l}
{y>-x+2} \\
{y<-x+4}
\end{array}\right.
$$

Grace B.

Numerade Educator

Problem 35

Graph the solutions of each system.
$$
\left\{\begin{array}{l}
{3 x+4 y \geq-7} \\
{2 x-3 y \geq 1}
\end{array}\right.
$$

Grace B.

Numerade Educator

Problem 36

Graph the solutions of each system.
$$
\left\{\begin{array}{l}
{3 x+y \leq 1} \\
{4 x-y \geq-8}
\end{array}\right.
$$

Grace B.

Numerade Educator

Problem 37

Graph the solutions of each system.
$$
\left\{\begin{array}{l}
{2 x+y<7} \\
{y>2-2 x}
\end{array}\right.
$$

Grace B.

Numerade Educator

Problem 38

Graph the solutions of each system.
$$
\left\{\begin{array}{l}
{2 x+y \geq 6} \\
{y \leq 2(2 x-3)}
\end{array}\right.
$$

Grace B.

Numerade Educator

Problem 39

Graph the solutions of each system.
$$
\left\{\begin{array}{l}
{2(x-2 y)>-6} \\
{3 x+y \geq 5}
\end{array}\right.
$$

Grace B.

Numerade Educator

Problem 40

Graph the solutions of each system.
$$
\left\{\begin{array}{l}
{2 x-3 y<0} \\
{2 x+3 y \geq 12}
\end{array}\right.
$$

Grace B.

Numerade Educator

Problem 41

Graph the solutions of each system.
$$
\left\{\begin{array}{l}
{3 x-y+4 \leq 0} \\
{3 y>-2 x-10}
\end{array}\right.
$$

Grace B.

Numerade Educator

Problem 42

Graph the solutions of each system.
$$
\left\{\begin{array}{l}
{3 x+2 y-12 \geq 0} \\
{x<-2+y}
\end{array}\right.
$$

Grace B.

Numerade Educator

Problem 43

Graph the solutions of each system.
$$
\left\{\begin{array}{l}
{x \geq-1} \\
{y \leq-x} \\
{x-y \leq 3}
\end{array}\right.
$$

Grace B.

Numerade Educator

Problem 44

Graph the solutions of each system.
$$
\left\{\begin{array}{l}
{y>-2.5} \\
{2 x-y \geq 2} \\
{x \leq 2}
\end{array}\right.
$$

Grace B.

Numerade Educator

Problem 45

Graph the solutions of each system.
$$
\left\{\begin{array}{l}
{x+y>0} \\
{y-x<-2}
\end{array}\right.
$$

Grace B.

Numerade Educator

Problem 46

Graph the solutions of each system.
$$
\left\{\begin{array}{l}
{y+2 x \leq 0} \\
{y \leq \frac{1}{2} x+2}
\end{array}\right.
$$

Grace B.

Numerade Educator

Problem 47

In part a, graph the solution of each system. Use your answer to part a to determine the solution of the system of equations in part b. (No new work is needed.)
a. $\left\{\begin{array}{l}{x+y>-1} \\ {y \geq x-3}\end{array}\right.$
b. $\left\{\begin{array}{l}{x+y=-1} \\ {y=x-3}\end{array}\right.$

Grace B.

Numerade Educator

Problem 48

In part a, graph the solution of each system. Use your answer to part a to determine the solution of the system of equations in part b. (No new work is needed.)
$$
\text { a. }\left\{\begin{array}{l}
{x-2 y \geq 6} \\
{y<-\frac{1}{2} x+1}
\end{array} \quad \text { b. }\left\{\begin{array}{l}
{x-2 y=6} \\
{y=-\frac{1}{2} x+1}
\end{array}\right.\right.
$$

Grace B.

Numerade Educator

Problem 49

Birds of Prey. Parts (a) and (b) of the illustration show the fields of vision for each eye of an owl. In part (c), shade the area where the fields of vision overlap-that is, the area that is seen by both eyes. (IMAGE CANNOT COPY)

Grace B.

Numerade Educator

Problem 50

Earth Science. Shade the area of the earth's surface that is north of the Tropic of Capricorn and south of the Tropic of Cancer. (IMAGE CANNOT COPY)

Grace B.

Numerade Educator

Problem 51

Graph each system of inequalities and give two possible solutions. See Example 5.
(GRAPH CANNOT COPY)
Buying Compact Discs. Melodic Music has compact discs on sale for either 10 dollars or 15 dollars. If a customer wants to spend at least 30 dollars but no more than 60 dollars on CDs, graph a system of inequalities showing the possible combinations of 10 dollars $\mathrm{CDs}$ $(x)$ and 15 dollars CDs $(y)$ that the customer can buy.

Check back soon!

Problem 52

Graph each system of inequalities and give two possible solutions. See Example 5.
(GRAPH CANNOT COPY)
Boats. Boatworks wholesales aluminum boats for 800 dollars and fiberglass boats for 600 dollars. Northland Marina wants to make a purchase totaling at least 2,400 dollars but no more than 4,800 dollars. Graph a system of inequalities showing the possible combinations of aluminum boats $(x)$ and fiberglass boats $(y)$ that can be ordered.

Check back soon!

Problem 53

Graph each system of inequalities and give two possible solutions. See Example 5.
(GRAPH CANNOT COPY)
Furniture. A distributor wholesales desk chairs for 150 dollars and side chairs for 100 dollars. Best Furniture wants its order to total no more than 900 dollars; Best also wants to order more side chairs than desk chairs. Graph a system of inequalities showing the possible combinations of desk chairs $(x)$ and side chairs $(y)$ that can be ordered.

Check back soon!

Problem 54

Graph each system of inequalities and give two possible solutions. See Example 5.
(GRAPH CANNOT COPY)
Air Ouality. J. Bolden Heating Company wants to order no more than 2,000 dollars worth of electronic air cleaners and humidifiers from a wholesaler that charges 500 dollars for air cleaners and 200 dollars for humidifiers. If Bolden wants more humidifiers than air cleaners, graph a system of inequalities showing the possible combinations of air cleaners $(x)$ and humidifiers $(y)$ that can be ordered.

Check back soon!

Problem 55

In some cameras, the image that you see in the viewfinder does not exactly match the image that will be recorded through the lens. To understand this difference, graph the solutions of the system
$$
\left\{\begin{array}{l}
{y \leq \frac{1}{4} x+2} \\
{y \geq-\frac{1}{4} x+2}
\end{array}\right.
$$
on the grid below. The shaded solution shows how the viewfinder image is slightly different from the lens image for extreme close-ups. (GRAPH CANNOT COPY)

Check back soon!

Problem 56

Pesticides. To get rid of fruit flies, helicopters sprayed an area of a city that can be described by $y \geq-2 x+1$ (within the city limits). Two weeks later, more spraying was ordered over the area described by $y \geq \frac{1}{4} x-4$ (within the city limits). Show the part of the city that was sprayed twice. (GRAPH CANNOT COPY)

Check back soon!

Problem 57

Explain the error. Graph: $\quad\left\{\begin{array}{l}{y>x} \\ {x+y<1}\end{array}\right.$

Grace B.

Numerade Educator

Problem 58

When a solution of a system of linear inequalities is graphed, what does the shading represent?

Grace B.

Numerade Educator

Problem 59

Describe how the graphs of the solutions of these systems are similar and how they differ.
$$
\left\{\begin{array}{l}
{x+y=4} \\
{x-y=4}
\end{array} \text { and } \quad\left\{\begin{array}{l}
{x+y \geq 4} \\
{x-y \geq 4}
\end{array}\right.\right.
$$

Grace B.

Numerade Educator

Problem 60

Explain when a system of inequalities will have no solutions.

Grace B.

Numerade Educator

Problem 61

Simplify each expression.
$$
8\left(\frac{3}{4} t\right)
$$

Grace B.

Numerade Educator

Problem 62

Simplify each expression.
$$
-\frac{2}{3}(3 w-6)
$$

Grace B.

Numerade Educator

Problem 63

Simplify each expression.
$$
-\frac{7}{16} x-\frac{3}{16} x
$$

Grace B.

Numerade Educator

Problem 64

Simplify each expression.
$$
60\left(\frac{3}{20} r-\frac{4}{15}\right)
$$

Grace B.

Numerade Educator

Problem 65

Graph the solutions of each system.
$$
\left\{\begin{array}{l}
{\frac{x}{3}-\frac{y}{2}<-3} \\
{\frac{x}{3}+\frac{y}{2}>-1}
\end{array}\right.
$$

Grace B.

Numerade Educator

Problem 66

Graph the solutions of each system.
$$
\left\{\begin{array}{l}
{3 x+y<-2} \\
{y>3(1-x)}
\end{array}\right.
$$

Grace B.

Numerade Educator

Problem 67

Graph the solutions of each system.
$$
\left\{\begin{array}{l}
{2 x+3 y \leq 6} \\
{3 x+y \leq 1} \\
{x \leq 0}
\end{array}\right.
$$

Grace B.

Numerade Educator

Problem 68

Graph the solutions of each system.
$$
\left\{\begin{array}{l}
{x \geq 0} \\
{y \geq 0} \\
{9 x+3 y \leq 18} \\
{3 x+6 y \leq 18}
\end{array}\right.
$$

Grace B.

Numerade Educator