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• How do I solve absolute value equations and inequalities?
2.2 Solve Absolute Value Equations and Inequalities
Example 1 Solve a simple absolute value equationSolve a simple absolute value equation
Solve . Graph the solution.63 xWrite original equation.63 xWrite equivalent equations.
___3 x ___3or x6 6
Solve for x._____x _____or x63 63Simplify.___x ___or x3 9
The solutions are ____ and ___. These are the values of x that are ___ units away from ___ on a number line.
3 96 3
l l l l l l l l l l l l l l l l l
6 units 6 units
2.2 Solve Absolute Value Equations and Inequalities
Checkpoint. Solve the equation. Then graph the solution Checkpoint. Solve the equation. Then graph the solution
5 1. x5x 5or x
l l l l l l l l l l l l l l l l l
5 units 5 units
25 2. x25 x 25or x
3x 7or x
l l l l l l l l l l l
2 units 2 units
2.2 Solve Absolute Value Equations and Inequalities
Example 2 Check for extraneous solutionsCheck for extraneous solutions
Solve . Check for extraneous solutions.xx 6104 Write original equation.
xx 6104 ___104 x ___104or xx6 x6 Expressions can
equal ____ or _____.x6 x6Subtract _____ from each side.
x4_____10 _____01or x2 x10
Solve for x.x___ x___or 5 1Check the apparent solutions to see if either is extraneous.
2.2 Solve Absolute Value Equations and Inequalities
Example 2 Check for extraneous solutionsCheck for extraneous solutions
Solve . Check for extraneous solutions.xx 6104 Check
xx 6104
__610__4?
xx 6104
__610__4?
5 5 1 1
______?
______?
30 30 6 6
______ 30 30 ______ 6 6The solution is ___. Reject ____ because it is an ___________ solution.
5 1 extraneous
2.2 Solve Absolute Value Equations and Inequalities
Checkpoint. Solve the equation. Check for extraneous Checkpoint. Solve the equation. Check for extraneous solutions. solutions.
1152 3. x1152 x 1152 x
5 56x2
2 2x 3
5 516x2
2 2x 8
11532?
11582?
1111 1111
2.2 Solve Absolute Value Equations and Inequalities
Checkpoint. Solve the equation. Check for extraneous Checkpoint. Solve the equation. Check for extraneous solutions. solutions.
xx 6183 4. xx 6183 xx 6183
x3 x3x318
3 3x 6
x3 x3x918
9 9x 2
661863?
261823?
3636 1212
2.2 Solve Absolute Value Equations and Inequalities
Rewrite the absolute value equation as two linear equations. Rewrite the absolute value equation as two linear equations.
23 1. x
Pg. 31 #1 and 17Pg. 31 #1 and 17
23 x 23or xSolve the equation. Solve the equation.
84 17. x84 x 84or x
4 4x 4
4 4x 12
2.2 Solve Absolute Value Equations and Inequalities
Example 3 Solve an inequality of the form .Solve an inequality of the form .cbax
The absolute value inequality is equivalent to.___52or ____52 xx 3
Solve . Then graph the solution.352 x
First Inequality Second InequalityWrite inequalities.___52 x ___52 x
3
3 3Subtract __ from each side.
5___2 x ___2 x8 2Divide each side by ___.2
___x ___x4 1
The solution are all real numbers less than ____ or greater than ____.
41
l l l l l l l l l l l l l l l l l
2.2 Solve Absolute Value Equations and Inequalities
Example 4 Solve an inequality of the form .Solve an inequality of the form .cbax Solve . Then graph the solution.5.45.1 x
Write inequality.5.45.1 x
The solution is between ____ and ____, inclusive.3 6
l l l l l l l l l l l l l l l l l
___5.1____ x5.4 5.4 Write equivalent compound inequality.
Add ____ to each expression.
5.1_______ x3 6
2.2 Solve Absolute Value Equations and Inequalities
Checkpoint. Solve the inequality. Then graph the solution. Checkpoint. Solve the inequality. Then graph the solution.
72 5. x
72 x 72or x2 2x 9
2 2x 5
l l l l l l l l l l l l l l l l l
2.2 Solve Absolute Value Equations and Inequalities
Checkpoint. Solve the inequality. Then graph the solution. Checkpoint. Solve the inequality. Then graph the solution.
914 6. x
9149 x1 1
1084 4
2
1 x4
4x 5.2
l l l l l l l l l
2.2 Solve Absolute Value Equations and Inequalities
Rewrite the absolute value inequality as a compound inequality. Rewrite the absolute value inequality as a compound inequality.
74 21. x
Pg. 32 #21, 24, 36, and 37Pg. 32 #21, 24, 36, and 37
7 4x 765 24. x65 x or 65 x
2.2 Solve Absolute Value Equations and Inequalities
Pg. 32 #21, 24, 36, and 37Pg. 32 #21, 24, 36, and 37Solve the inequality. Solve the inequality.
31 36. x31x 31or x
1 1x 2
1 1x 4
2or 4 xx
2.2 Solve Absolute Value Equations and Inequalities
Pg. 32 #21, 24, 36, and 37Pg. 32 #21, 24, 36, and 37Solve the inequality. Solve the inequality.
932 37. x 9 32 x
336
3x2 12
9
2 2 2
3 x 6
2.2 Solve Absolute Value Equations and Inequalities
Pg. 31, 2.2 #1-43
Textbook pg 30 #1-15, 16-28
