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Difficulty: HardAbsolute Value

Determine the solutions for xx in the given absolute value equation.

Answer:Given the equation 2∣5−x∣−4=102|5 - x| - 4 = 10, the smaller value of xx is 【-2】 and the larger value of xx is 【12】.

Answer

The smaller value is -2 and the larger value is 12.
To solve 2∣5−x∣−4=102|5 - x| - 4 = 10, first isolate the absolute value term. Adding 4 to both sides yields 2∣5−x∣=142|5 - x| = 14. Dividing both sides by 2 gives ∣5−x∣=7|5 - x| = 7. The absolute value of an expression equals a positive number when the expression itself equals that positive number or its negative counterpart. Therefore, we set up two cases: 5−x=75 - x = 7 and 5−x=−75 - x = -7. Solving the first case gives −x=2-x = 2, so x=−2x = -2. Solving the second case gives −x=−12-x = -12, so x=12x = 12. Comparing the two solutions, the smaller value is −2-2 and the larger value is 1212.

Step-by-Step Solution

1
Isolate the absolute value expression.
Add 4 to both sides to get 2∣5−x∣=142|5 - x| = 14, then divide by 2 to get ∣5−x∣=7|5 - x| = 7.
The absolute value must be isolated before separating the equation into cases.
2
Set up the positive case equation.
5−x=75 - x = 7
The first case assumes the expression inside the absolute value is positive or zero.
3
Solve the positive case.
Subtract 5 to get −x=2-x = 2, which means x=−2x = -2.
This finds the first potential value for xx.
4
Set up the negative case equation.
5−x=−75 - x = -7
The second case accounts for the expression inside the absolute value being negative.
5
Solve the negative case.
Subtract 5 to get −x=−12-x = -12, which means x=12x = 12.
This finds the second potential value for xx.

Key Concept

Solving Linear Absolute Value Equations
Estimated Time:1m 30s
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Topics

Absolute Value

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