Holt Algebra 1

2-4Solving Equations with Variables on Both Sides

Group Quiz

Solve each equation.

Bellringer: Finish this group quiz from yesterday.

1. 15 = |x| 2. 2|x – 7| = 14

3. |x + 1|– 9 = –9 4. |5 + x| – 3 = –2

5. 7 + |x – 8| = 6

–15, 15 0, 14

–1–6, –4

no solution

Holt Algebra 1

2-4Solving Equations with Variables on Both Sides2-4 Agenda

Holt Algebra 1

•Bellringer – Group QuizBellringer – Group Quiz

•Solving Absolute-Value EquationsSolving Absolute-Value Equations

•Book ProblemsBook Problems

•Solving in words activitySolving in words activity

•WorksheetWorksheet

Holt Algebra 1

2-4Solving Equations with Variables on Both Sides

Recall that the absolute-value of a number is that number’s distance from zero on a number

line. For example, |–5| = 5 and |5| = 5.

5 4 3 2 0 1 2 3 4 56 1 6

5 units

To write this statement using algebra, you would write |x| = 5. This equation asks, “What values of x have an absolute value of 5?” The solutions are 5

and –5. Notice this equation has two solutions.

5 units

Holt Algebra 1

2-4Solving Equations with Variables on Both Sides

Holt Algebra 1

2-4Solving Equations with Variables on Both Sides

Additional Example 1A: Solving Absolute-Value Equations

Solve the equation.

|x| = 12|x| = 12

Case 1 x = 12

Case 2 x = –12

The solutions are {12, –12}.

Think: What numbers are 12 units from 0?

Rewrite the equation as two cases.

12 units 12 units

10 8 6 4 0 2 4 6 8 1012 2 12•••

Holt Algebra 1

2-4Solving Equations with Variables on Both Sides

Additional Example 1A: Solving Absolute-Value Equations

Solve the equation.

|x| = 9|x| = 9

Case 1 x = 9

Case 2 x = –9

The solutions are {9, –9}.

Think: What numbers are 9 units from 0?

Rewrite the equation as two cases.

9 units 9 units

10 8 6 4 0 2 4 6 8 1012 2 12•••

Holt Algebra 1

2-4Solving Equations with Variables on Both Sides

Solve the equation.

Check It Out! Example 1b

8 =|x 2.5| Think: What numbers are

8 units from 0?

Case 18 = x 2.5

+2.5 +2.5

10.5 = x

+2.5 +2.55.5 = x

Case 2 8 = x 2.5

Rewrite the equations as two cases.

The solutions are {10.5, –5.5}.

8 =|x 2.5|

Since 2.5 is subtracted from x add 2.5 to both sides of each equation.

Holt Algebra 1

2-4Solving Equations with Variables on Both Sides

Solve the equation.

Check It Out! Example 1b

7 =|x 4|

Case 17 = x 4

+4 +4

11 = x

+4 +4-3 = x

Case 2 7 = x 4

The solutions are {11, –3}.

7 =|x 4|

Holt Algebra 1

2-4Solving Equations with Variables on Both Sides

Solve the equation.

Check It Out! Example 1a

|x| – 3 = 4|x| – 3 = 4

+ 3 +3|x| = 7

Case 1 x = 7

Case 2 x = –7

The solutions are {7, –7}.

Since 3 is subtracted from |x|, add 3 to both sides.

Think: What numbers are 7 units from 0?

Rewrite the equation as two cases.

Holt Algebra 1

2-4Solving Equations with Variables on Both Sides

Additional Example 1B: Solving Absolute-Value Equations

3|x + 7| = 24

|x + 7| = 8

The solutions are {1, –15}.

Case 1 x + 7 = 8

Case 2 x + 7 = –8

– 7 –7 – 7 – 7x = 1 x = –15

Since |x + 7| is multiplied by 3, divide both sides by 3 to undo

the multiplication.

Think: What numbers are 8 units from 0?

Rewrite the equations as two cases. Since 7 is added to x subtract 7 from both sides

of each equation.

Solve the equation.

Holt Algebra 1

2-4Solving Equations with Variables on Both Sides

Check It Out! Example 2b

Solve the equation.

6 + |x 4| = 6

6 + |x 4| = 6+6 +6

|x 4| = 0

x 4 = 0+ 4 +4

x = 4

Since –6 is added to |x 4|, add 6 to both

sides.

There is only one case. Since 4 is subtracted from x, add 4 to both sides to

undo the addition.

Holt Algebra 1

2-4Solving Equations with Variables on Both Sides

2nd Problem

Solve the equation.

7 + |x 3| = -2

7 + |x 4| = 2+7 +7

|x 4| = 5

x 4 = -5+ 4 +4

x = -1

x 4 = 5

x 4 = 5+ 4 +4

x = 9

Holt Algebra 1

2-4Solving Equations with Variables on Both Sides

Additional Example 2B: Special Cases of Absolute-Value Equations

Solve the equation.

3 + |x + 4| = 0

3 + |x + 4| = 03 3

|x + 4| = 3 Absolute value cannot be negative.

This equation has no solution.

Holt Algebra 1

2-4Solving Equations with Variables on Both Sides

Remember!

Absolute value must be nonnegative because it represents a distance.

Holt Algebra 1

2-4Solving Equations with Variables on Both Sides

Check It Out! Example 2a

Solve the equation.

2 |2x 5| = 7

2 |2x 5| = 7 2 2

|2x 5| = 5

Since 2 is added to –|2x – 5|, subtract 2 from both sides to

undo the addition.

Absolute value cannot be negative.

|2x 5| = 5

This equation has no solution.

Since |2x – 5| is multiplied by negative 1, divide both sides

by negative 1.

Holt Algebra 1

2-4Solving Equations with Variables on Both Sides

Word Problems

• Troy’s car can go 24 miles on one gallon of gasoline. However, his gas mileage can vary from this value by 2 miles per gallon depending on where he drives.

• Write an absolute-value equation that you can use to find the minimum and maximum gas mileage.│x – 24 │= 2

• Solve the equation to find the minimum and maximum gas mileage.

x – 24 = 2 and x – 24 = -2

+ 24 +24 + 24 + 24

x = 26 and x = 22

Holt Algebra 1

2-4Solving Equations with Variables on Both Sides

• Book problems

• Solving Equations with Words activity.

• Update your flipbooks! Most of you have not done this lately. You are getting a grade for this.

• Worksheet