Question

For what values of $x$ will $\sqrt{(3 x-8)^{2}}=3 x-8$ ? Explain how you determined your answer.

Name: Problem 116, Chapter 7: Intermediate Algebra for College Students
Uploaded: 2020-07-15T19:05:31-08:00
Duration: 6 min 42 s
Channel: Vishal Parmar
Description: For what values of x will √((3 x-8)^2)=3 x-8 ? Explain how you determined your answer.

   For what values of $x$ will $\sqrt{(3 x-8)^{2}}=3 x-8$ ? Explain how you determined your answer.

Intermediate Algebra for College Students

Allen R. Angel;… 10th Edition

Chapter 7, Problem 116

Instant Answer

Solved by Expert Vishal Parmar

07/15/2020

Step 1

So, we have two cases to consider: Case 1: $3x - 8 \ge 0$ In this case, we have $\sqrt{(3x - 8)^2} = 3x - 8$. Squaring both sides, we get $(3x - 8)^2 = (3x - 8)^2$. This equation is true for all $x$ in this case, so we need to find the values of $x$ that satisfy Show more…

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Key Concepts

Square Root Properties

The square root function always returns the non-negative result for any non-negative input. This principle is vital when simplifying expressions like ?((3x - 8)²) because it leads to the realization that ?((3x - 8)²) equals the absolute value |3x - 8|.

Absolute Value

The absolute value function measures the magnitude of a number regardless of its sign. Recognizing that ?((3x - 8)²) = |3x - 8| allows us to see that the original equation becomes |3x - 8| = 3x - 8, which naturally holds only when 3x - 8 is non-negative.

Domain Restrictions

When solving equations involving absolute values or square roots, it is crucial to consider the domain of the expressions involved. In this example, since |3x - 8| equals 3x - 8 only when 3x - 8 is greater than or equal to zero, this leads to the condition x ? 8/3.

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