Section 2.8 – Absolute Value Equations and Inequalities Recall ...

Section 2.8 – Absolute Value Equations and Inequalities
Recall: Absolute Value of a point on the number line is the distance of that point to the origin
‘0’.
| 4 | =
| − 5 | =
| 5(
2
− 4)
| =
Solving Absolute Value Equations
We want to solve equations of this form: x = C
1) If C is positive: x = C has two solutions; x = C or x = −C
2) If C is negative: x = C has no solution
3) If C = 0 : x = 0 has one solution: x = 0
Example 1: Solve x = 8
Example 2: Solve x = −2
Remember: “absolute value = negative number” has ____ solution.
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How to solve absolute value equations
1. Isolate the absolute value expression on one side and a number on the other by adding or
subtracting first, then multiplying or dividing. Get: Expression = Number
2. If the resulting equation is absolute value equals a positive number, rewrite into the two
equivalent equations. These equations do NOT have absolute value signs!
3. If the resulting equation is absolute value equals 0, set the expression in the absolute
value equal to 0 and solve.
4. If the resulting equation is absolute value equals a negative number, there is no solution,
you can stop here and state “no solution”
Example 3: Solve x + 4 = 9.
Example 4: Solve 2x − 4 = 8 .
Example 5: Solve 4 x − 6 = 20.
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Example 6: Solve 2 2x
+ 1 + 5 = 21.
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Example 7: Solve 2x
− 6 = 4 .
4
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Example 8: Solve 20 + 2x + 4 = 15
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Solving Absolute Value Inequalities
If C is positive, then:
a) x < C is equivalent to − C < x < C .
b) x ≤ C is equivalent to − C ≤ x ≤ C .
c) x > C is equivalent to x < −C
or x > C .
d) x ≥ C is equivalent to x ≤ −C
or x ≥ C .
Special Cases:
If C is negative, then
If C = 0, then
a) The inequalities of the form x < C or x ≤ C have no solution.
b) The solution of the inequalities x > C or x ≥ C is all real numbers.
a) The inequalities of the form x < 0 has no solution.
b) The solution of the inequality x = 0 is x = 0.
c) The solution of the inequality x > 0 is all real numbers except 0.
d) The solution of the inequality x ≥ 0 is all real numbers.
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How to solve absolute value inequalities:
1. Rearrange the equation so that you have the form:
Absolute value inequality sign number
2. Write the equivalent inequality relation or relations by the rules on the last page. Note
that these equivalent inequalities no longer have absolute value signs!
3. Solve each inequality for x and express the answer in interval notation
4. If the equation is equivalent to “x > 3 or x < -3” type inequality, use the union symbol
between the two intervals in your answer.
Example 9: Solve the inequality and express your answer in interval notation.
a. x ≤ 5
b. x < 5
c. x ≥ 5
d. x
> 5
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Example 10: Solve the inequality and express your answer in interval notation.
x + 4 <
5
Example 11: Solve the inequality and express your answer in interval notation.
4 2x
+ 1 ≤ 28
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Example 12: Solve the inequality and express your answer in interval notation.
2 2x
−1
+ 1 > 7
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Example 13: Solve the inequality and express your answer in interval notation.
2 10
x
− 4 ≥
3 3
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Example 14: Solve the inequality and express your answer in interval notation.
x + 1
< 2
5
Example 15: Solve the inequality and express your answer in interval notation.
−5 ≤ 9x + 2
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Example 16: Solve 7 + 2 + 3x ≤ 4 .
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