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