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A.3 Equations, Inequalities, and Problem Solving

A.3 Equations, Inequalities, and Problem Solving

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A10

Appendix A

Review of Elementary Algebra Topics

EXAMPLE 2

Solving a Linear Equation in Nonstandard Form

Solve 5x ϩ 4 ϭ 3x Ϫ 8.

Solution

5x ϩ 4 ϭ 3x Ϫ 8

5x Ϫ 3x ϩ 4 ϭ 3x Ϫ 3x Ϫ 8

2x ϩ 4 ϭ Ϫ8

2x ϩ 4 Ϫ 4 ϭ Ϫ8 Ϫ 4

Write original equation.

Subtract 3x from each side.

Combine like terms.

Subtract 4 from each side.

2x ϭ Ϫ12

Combine like terms.

2x Ϫ12

ϭ

2

2

Divide each side by 2.

x ϭ Ϫ6

Simplify.

The solution is x ϭ Ϫ6. Check this in the original equation.

Linear equations often contain parentheses or other symbols of grouping. In

most cases, it helps to remove symbols of grouping as a first step in solving an

equation. This is illustrated in Example 3.

EXAMPLE 3

Solving a Linear Equation Involving Parentheses

Solve 2͑x ϩ 4͒ ϭ 5͑x Ϫ 8͒.

Solution

2͑x ϩ 4͒ ϭ 5͑x Ϫ 8͒

2x ϩ 8 ϭ 5x Ϫ 40

2x Ϫ 5x ϩ 8 ϭ 5x Ϫ 5x Ϫ 40

Ϫ3x ϩ 8 ϭ Ϫ40

Study Tip

Recall that when finding the least

common multiple of a set of

numbers, you should first consider

all multiples of each number. Then,

you should choose the smallest of

the common multiples of the

numbers.

Ϫ3x ϩ 8 Ϫ 8 ϭ Ϫ40 Ϫ 8

Write original equation.

Distributive Property

Subtract 5x from each side.

Combine like terms.

Subtract 8 from each side.

Ϫ3x ϭ Ϫ48

Combine like terms.

Ϫ3x Ϫ48

ϭ

Ϫ3

Ϫ3

Divide each side by Ϫ3.

x ϭ 16

Simplify.

The solution is x ϭ 16. Check this in the original equation.

To solve an equation involving fractional expressions, find the least common

multiple (LCM) of the denominators and multiply each side by the LCM.

Section A.3

Equations, Inequalities, and Problem Solving

Solving a Linear Equation Involving Fractions

EXAMPLE 4

Solve

A11

x

3x

ϩ

ϭ 2.

3

4

Solution

12

12 и

΂3x ϩ 3x4 ΃ ϭ 12͑2͒

Multiply each side of original

equation by LCM 12.

x

3x

ϩ 12 и

ϭ 24

3

4

Distributive Property

4x ϩ 9x ϭ 24

Clear fractions.

13x ϭ 24

Combine like terms.

24

13

Divide each side by 13.

The solution is x ϭ 24

13 . Check this in the original equation.

To solve an equation involving an absolute value, remember that the expression

inside the absolute value signs can be positive or negative. This results in two

separate equations, each of which must be solved.

Solving an Equation Involving Absolute Value

EXAMPLE 5

Խ

Խ

Solve 4x Ϫ 3 ϭ 13.

Solution

Խ4x Ϫ 3Խ ϭ 13

Write original equation.

4x Ϫ 3 ϭ Ϫ13 or 4x Ϫ 3 ϭ 13

4x ϭ Ϫ10

xϭϪ

5

2

4x ϭ 16

xϭ4

Equivalent equations

Divide each side by 4.

The solutions are x ϭ Ϫ 52 and x ϭ 4. Check these in the original equation.

Inequalities

The simplest type of inequality is a linear inequality in one variable. For instance,

2x ϩ 3 > 4 is a linear inequality in x. The procedures for solving linear

inequalities in one variable are much like those for solving linear equations, as

described on page A9. The exception is that when each side of an inequality is

multiplied or divided by a negative number, the direction of the inequality

symbol must be reversed.

A12

Appendix A

Review of Elementary Algebra Topics

Solving a Linear Inequality

EXAMPLE 6

Solve and graph the inequality Ϫ5x Ϫ 7 > 3x ϩ 9.

Solution

Ϫ5x Ϫ 7 > 3x ϩ 9

Write original inequality.

Ϫ8x Ϫ 7 > 9

Ϫ8x > 16

x < −2

−3

−2

−1

Divide each side by Ϫ8 and reverse the

direction of the inequality symbol.

x < Ϫ2

x

−4

Subtract 3x from each side.

The solution set in interval notation is ͑Ϫ ϱ, Ϫ2͒ and in set notation is

ͭx x < Ϫ2ͮ. The graph of the solution set is shown in Figure A.8.

0

Խ

Figure A.8

Two inequalities joined by the word and or the word or constitute a compound

inequality. Sometimes it is possible to write a compound inequality as a double

inequality. For instance, you can write Ϫ3 < 6x Ϫ 1 and 6x Ϫ 1 < 3 more

simply as Ϫ3 < 6x Ϫ 1 < 3. A compound inequality formed by the word and is

called conjunctive and may be rewritten as a double inequality. A compound

inequality joined by the word or is called disjunctive and cannot be rewritten as

a double inequality.

EXAMPLE 7

Solving a Conjunctive Inequality

Solve and graph the inequality 2x ϩ 3 Ն 4 and 3x Ϫ 8 < Ϫ2.

Solution

2x ϩ 3 Ն 4

1

2

1

2

≤ x<2

x

−1

0

1

2

3

Study Tip

2x Ն 1

3x < 6

1

2

x < 2

x Ն

x > 5

x>5

Figure A.10

2

3

4

5

Յ x < 2ͮ.

Solving a Disjunctive Inequality

Solution

6

or

Ϫ6x ϩ 1 Ն Ϫ5

Ϫ6x Ն Ϫ6

x Յ 1

x

1

1

2

Solve and graph the inequality x Ϫ 8 > Ϫ3 or Ϫ6x ϩ 1 Ն Ϫ5.

x Ϫ 8 > Ϫ3

0

Խ

The solution set in interval notation is ͓2, 2͒ and in set notation is ͭ x

The graph of the solution set is shown in Figure A.9.

EXAMPLE 8

Recall that the word or is

represented by the symbol ʜ,

−1

3x Ϫ 8 < Ϫ2

1

Figure A.9

x≤1

and

The solution set in interval notation is ͑Ϫ ϱ, 1͔ ʜ ͑5, ϱ͒ and in set notation is

ͭx x > 5 or x Յ 1ͮ. The graph of the solution set is shown in Figure A.10.

Խ

Section A.3

Equations, Inequalities, and Problem Solving

A13

To solve an absolute value inequality, use the following rules.

Solving an Absolute Value Inequality

Let x be a variable or an algebraic expression and let a be a real number

such that a > 0.

1. The solutions of x < a are all values of x that lie between Ϫa and a.

x < a if and only if Ϫa < x < a

ԽԽ

ԽԽ

ԽԽ

2. The solutions of x > a are all values of x that are less than Ϫa or

greater than a.

x > a if and only if x < Ϫa or x > a

ԽԽ

These rules are also valid if < is replaced by Յ and > is replaced by Ն .

Solving Absolute Value Inequalities

EXAMPLE 9

Solve and graph each inequality.

a. 4x ϩ 3 > 9

b. 2x Ϫ 7 Յ 1

Խ

Խ

Խ

Խ

Solution

a. 4x ϩ 3 > 9

Խ

Խ

Write original inequality.

4x ϩ 3 < Ϫ9 or

4x < Ϫ12

x < Ϫ3

4x ϩ 3 > 9

4x > 6

x > 32

Equivalent inequalities

Subtract 3 from each side.

Divide each side by 4.

The solution set consists of all real numbers that are less than Ϫ3 or greater

than 32. The solution set in interval notation is ͑Ϫ ϱ, Ϫ3͒ ʜ ͑32, ϱ͒ and in set

notation is ͭ x x < Ϫ3 or x > 32ͮ. The graph is shown in Figure A.11.

Խ

x < −3

x > 32

3

2

x

−4 −3 −2 −1

0

1

2

3

Figure A.11

Խ

Խ

b. 2x Ϫ 7 Յ 1

Write original inequality.

Ϫ1 Յ 2x Ϫ 7 Յ 1

3≤ x≤4

x

1

2

Figure A.12

3

4

5

Equivalent double inequality

6 Յ 2x Յ 8

Add 7 to all three parts.

3 Յ x Յ 4

Divide all three parts by 2.

The solution set consists of all real numbers that are greater than or equal to 3

and less than or equal to 4. The solution set in interval notation is ͓3, 4͔ and in

set notation is ͭx 3 Յ x Յ 4ͮ. The graph is shown in Figure A.12.

Խ

A14

Appendix A

Review of Elementary Algebra Topics

Problem Solving

Algebra is used to solve word problems that relate to real-life situations. The

following guidelines summarize the problem-solving strategy that you should use

when solving word problems.

Guidelines for Solving Word Problems

1. Write a verbal model that describes the problem.

2. Assign labels to fixed quantities and variable quantities.

3. Rewrite the verbal model as an algebraic equation using the assigned

labels.

4. Solve the resulting algebraic equation.

5. Check to see that your solution satisfies the original problem as stated.

EXAMPLE 10

Finding the Percent of Monthly Expenses

Your family has an annual income of \$77,520 and the following monthly expenses:

mortgage (\$1500), car payment (\$510), food (\$400), utilities (\$325), and credit

cards (\$300). The total expenses for one year represent what percent of your

family’s annual income?

Solution

The total amount of your family’s monthly expenses is

1500 ϩ 510 ϩ 400 ϩ 325 ϩ 300 ϭ \$3035.

The total monthly expenses for one year are

3035 и 12 ϭ \$36,420.

Verbal

Model:

Expenses ϭ Percent

и

Income

Labels:

Expenses ϭ 36,420

Percent ϭ p

Income ϭ 77,520

Equation:

36,420 ϭ p и 77,520

Original equation

36,420

ϭp

77,520

Divide each side by 77,520.

0.470 Ϸ p

(dollars)

(in decimal form)

(dollars)

Use a calculator.

Your family’s total expenses for one year are approximately 0.470 or 47.0% of

Section A.3

EXAMPLE 11

A15

Equations, Inequalities, and Problem Solving

Geometry: Similar Triangles

To determine the height of the Aon Center Building (in Chicago), you measure

the shadow cast by the building and find it to be 142 feet long, as shown in Figure

A.13. Then you measure the shadow cast by a four-foot post and find it to be

6 inches long. Estimate the building’s height.

x ft

Solution

To solve this problem, you use a property from geometry that states that the ratios

of corresponding sides of similar triangles are equal.

48 in.

Height of building

Verbal

Model:

Height of post

ϭ

6 in.

142 ft

Not drawn to scale

Height of building ϭ x

Length of building’s shadow ϭ 142

Height of post ϭ 4 feet ϭ 48 inches

Length of post’s shadow ϭ 6

Labels:

Figure A.13

x

48

ϭ

142

6

Proportion:

x и 6 ϭ 142 и 48

x ϭ 1136

(feet)

(feet)

(inches)

(inches)

Original proportion

Cross-multiply

Divide each side by 6.

So, you can estimate the Aon Center Building to be 1136 feet high.

EXAMPLE 12

Geometry: Dimensions of a Room

A rectangular kitchen is twice as long as it is wide, and its perimeter is 84 feet.

Find the dimensions of the kitchen.

w

l

Figure A.14

Solution

For this problem, it helps to sketch a diagram, as shown in Figure A.14.

Verbal

Model:

Labels:

2

и

Length ϩ 2 и Width ϭ Perimeter

Length ϭ l ϭ 2w

Width ϭ w

Perimeter ϭ 84

Equation: 2͑2w͒ ϩ 2w ϭ 84

Study Tip

For more review on equations and

inequalities, refer to Chapter 3.

6w ϭ 84

w ϭ 14

(feet)

(feet)

(feet)

Original equation

Combine like terms.

Divide each side by 6.

Because the length is twice the width, you have l ϭ 2w ϭ 2͑14͒ ϭ 28. So, the

dimensions of the room are 14 feet by 28 feet.

A16

Appendix A

Review of Elementary Algebra Topics

᭤ The Rectangular Coordinate System

᭤ Graphs of Equations

A.4 Graphs and Functions

᭤ Functions

᭤ Slope and Linear Equations

The Rectangular Coordinate System

᭤ Graphs of Linear Inequalities

You can represent ordered pairs of real numbers by points in a plane. This plane

is called a rectangular coordinate system. A rectangular coordinate system is

formed by two real lines, the x-axis (horizontal line) and the y-axis (vertical line),

intersecting at right angles. The point of intersection of the axes is called the

origin, and the axes divide the plane into four regions called quadrants.

Each point in the plane corresponds to an ordered pair ͑x, y͒ of real numbers

x and y, called the coordinates of the point. The x-coordinate tells how far to the

left or right the point is from the vertical axis, and the y-coordinate tells how far

up or down the point is from the horizontal axis, as shown in Figure A.15.

y

4

3

2

y-axis

y-coordinate

1

−4 −3 −2 −1

−1

x-axis −2

(3, 2)

x-coordinate

−3

−4

x

1

2

3

4

Origin

Figure A.15

y

EXAMPLE 1

4

C

B

3

2

Determine the coordinates of each of the points shown in Figure A.16, and then

determine the quadrant in which each point is located.

D

1

− 4 − 3 −2 −1

−1

E

−2

−3

−4

Figure A.16

x

1

2

A

3

4

Finding Coordinates of Points

Solution

Point A lies two units to the right of the vertical axis and one unit below the

horizontal axis. So, point A must be given by ͑2, Ϫ1͒. The coordinates of the

other four points can be determined in a similar way. The results are as follows.

Point

A

B

C

D

E

Coordinates

͑2, Ϫ1͒

͑4, 3͒

͑Ϫ1, 3͒

͑0, 2͒

͑Ϫ3, Ϫ2͒

IV

I

II

None

III

Section A.4

A17

Graphs and Functions

Graphs of Equations

The solutions of an equation involving two variables can be represented by points

on a rectangular coordinate system. The graph of an equation is the set of all

points that are solutions of the equation.

The simplest way to sketch the graph of an equation is the point-plotting

method. With this method, you construct a table of values consisting of several

solution points of the equation, plot these points, and then connect the points with

a smooth curve or line.

Sketching the Graph of an Equation

EXAMPLE 2

Sketch the graph of y ϭ x 2 Ϫ 2.

Solution

Begin by choosing several x-values and then calculating the corresponding

y-values. For example, if you choose x ϭ Ϫ2, the corresponding y-value is

y ϭ x2 Ϫ 2

Original equation

y ϭ ͑Ϫ2͒ Ϫ 2

Substitute Ϫ2 for x.

y ϭ 4 Ϫ 2 ϭ 2.

Simplify.

2

Then, create a table using these values, as shown below.

x

y‫؍‬

x2

؊2

Ϫ2

Ϫ1

0

1

2

3

2

Ϫ1

Ϫ2

Ϫ1

2

7

͑Ϫ1, Ϫ1͒

͑0, Ϫ2͒

͑1, Ϫ1͒

͑2, 2͒

͑3, 7͒

͑Ϫ2, 2͒

Solution point

Next, plot the solution points, as shown in Figure A.17. Finally, connect the

points with a smooth curve, as shown in Figure A.18.

y

y

(3, 7)

6

6

4

4

2

2

y = x2 − 2

(−2, 2)

−4

−2

(−1, − 1)

Figure A.17

(2, 2)

x

2

(1, −1)

(0, −2)

4

−4

−2

Figure A.18

x

2

4

A18

Appendix A

Review of Elementary Algebra Topics

y

(−1, 3)

(−7, 3)

(− 2, 2)

Խ

5

Solution

Begin by creating a table of values, as shown below. Plot the solution points, as

shown in Figure A.19. It appears that the points lie in a “V-shaped” pattern, with

the point ͑Ϫ4, 0͒ lying at the bottom of the “V.” Following this pattern,

connect the points to form the graph shown in Figure A.20.

2

(− 3, 1) 1

(−5, 1)

x

− 7 − 6 −5 −4 −3 −2 −1

−1

(− 4, 0)

Խ

Sketch the graph of y ϭ x ϩ 4 .

4

3

(− 6, 2)

Sketching the Graph of an Equation

EXAMPLE 3

6

1

−2

x

Figure A.19

Խ

Ϫ7

Ϫ6

Ϫ5

Ϫ4

Ϫ3

Ϫ2

Ϫ1

3

2

1

0

1

2

3

Խ

y‫ ؍‬x14

y

Solution

point

6

͑Ϫ7, 3͒ ͑Ϫ6, 2͒ ͑Ϫ5, 1͒ ͑Ϫ4, 0͒ ͑Ϫ3, 1͒ ͑Ϫ2, 2͒ ͑Ϫ1, 3͒

5

y= x+4

4

Intercepts of a graph are the points at which the graph intersects the x- or

y-axis. To find x-intercepts, let y ϭ 0 and solve the equation for x. To find

y-intercepts, let x ϭ 0 and solve the equation for y.

3

2

1

x

− 7 − 6 −5 −4 −3 −2 −1

−1

1

EXAMPLE 4

−2

Finding the Intercepts of a Graph

Find the intercepts and sketch the graph of y ϭ 3x ϩ 4.

Figure A.20

Solution

To find any x-intercepts, let y ϭ 0 and solve the resulting equation for x.

y

y-intercept:

(0, 4)

8

x-intercept:

4

6

(− 43 , 0)

Write original equation.

0 ϭ 3x ϩ 4

Let y ϭ 0.

4

Ϫ ϭx

3

y = 3x + 4

Solve equation for x.

To find any y-intercepts, let x ϭ 0 and solve the resulting equation for y.

x

y ϭ 3x ϩ 4

Write original equation.

−4

y ϭ 3͑0͒ ϩ 4

Let x ϭ 0.

−6

yϭ4

− 8 −6 −4

2

−8

Figure A.21

y ϭ 3x ϩ 4

4

6

8

Solve equation for y.

So, the x-intercept is ͑

0͒ and the y-intercept is ͑0, 4͒. To sketch the graph of

the equation, create a table of values (including intercepts), as shown below. Then

plot the points and connect them with a line, as shown in Figure A.21.

Ϫ 43,

x

Ϫ3

Ϫ2

Ϫ 43

Ϫ1

0

1

y ‫ ؍‬3x 1 4

Ϫ5

Ϫ2

0

1

4

7

͑Ϫ3, Ϫ5͒

͑Ϫ2, Ϫ2͒

͑0, 4͒

͑1, 7͒

Solution point

͑Ϫ 43, 0͒

͑Ϫ1, 1͒

Section A.4

A19

Graphs and Functions

Functions

A relation is any set of ordered pairs, which can be thought of as (input, output).

A function is a relation in which no two ordered pairs have the same first

component and different second components.

Testing Whether Relations Are Functions

EXAMPLE 5

Decide whether the relation represents a function.

a.

b. Input: 2, 5, 7

a

1

Output: 1, 2, 3

ͭ͑2, 1͒, ͑5, 2͒, ͑7, 3͒ͮ

2

b

3

c

4

Input

Output

Solution

a. This diagram does not represent a function. The first component a is paired

with two different second components, 1 and 2.

b. This set of ordered pairs does represent a function. No first component has two

different second components.

The graph of an equation represents y as a function of x if and only if no

vertical line intersects the graph more than once. This is called the Vertical Line Test.

Using the Vertical Line Test for Functions

EXAMPLE 6

Use the Vertical Line Test to determine whether y is a function of x.

y

y

a.

b.

3

3

2

2

1

1

x

x

−1

−1

1

2

3

4

5

−3 −2 −1

−1

−2

−2

−3

−3

1

2

3

Solution

a. From the graph, you can see that a vertical line intersects more than one point

on the graph. So, the relation does not represent y as a function of x.

b. From the graph, you can see that no vertical line intersects more than one point

on the graph. So, the relation does represent y as a function of x.

A20

Appendix A

Review of Elementary Algebra Topics

Slope and Linear Equations

The graph in Figure A.21 on page A18 is an example of a graph of a linear

equation. The equation is written in slope-intercept form, y ϭ mx ϩ b, where m

is the slope and ͑0, b͒ is the y-intercept. Linear equations can be written in other

forms, as shown below.

Forms of Linear Equations

1. General form: ax ϩ by ϩ c ϭ 0

2. Slope-intercept form: y ϭ mx ϩ b

3. Point-slope form: y Ϫ y1 ϭ m͑x Ϫ x1͒

The slope of a nonvertical line is the number of units the line rises or falls

vertically for each unit of horizontal change from left to right. To find the slope

m of the line through ͑x1, y1͒ and ͑x2, y2͒, use the following formula.

y2 Ϫ y1 Change in y

ϭ

x2 Ϫ x1 Change in x

EXAMPLE 7

Find the slope of the line passing through ͑3, 1͒ and ͑Ϫ6, 0͒.

y

m=

3

1

9

2

(−6, 0)

9

−2

−3

Figure A.22

Finding the Slope of a Line Through Two Points

(3, 1)

1

x

Solution

Let ͑x1, y1͒ ϭ ͑3, 1͒ and ͑x2, y2͒ ϭ ͑Ϫ6, 0͒. The slope of the line through these

points is

4

y2 Ϫ y1

0Ϫ1

Ϫ1 1

ϭ

ϭ

ϭ .

x2 Ϫ x1 Ϫ6 Ϫ 3 Ϫ9 9

The graph of the line is shown in Figure A.22.

You can make several generalizations about the slopes of lines.

Slope of a Line

1. A line with positive slope ͑m > 0͒ rises from left to right.

2. A line with negative slope ͑m < 0͒ falls from left to right.

3. A line with zero slope ͑m ϭ 0͒ is horizontal.

4. A line with undefined slope is vertical.

5. Parallel lines have equal slopes: m1 ϭ m2

6. Perpendicular lines have negative reciprocal slopes: m1 ϭ Ϫ

1

m2

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