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Question Solve the absolute value equation ∣3x−1∣=10|3x-1| = 10 for the real number value(s) of xx.

Studdy Solution

STEP 1

Assumptions
1. We are given the absolute value equation ∣3x−1∣=10|3x - 1| = 10.
2. We need to solve for the variable xx.

STEP 2

Understand the definition of absolute value. The absolute value of a number is the distance of that number from 0 on the real number line, without considering direction. The absolute value of a number is always non-negative.

STEP 3

Set up two separate equations to solve the absolute value equation. Since ∣a∣=b|a| = b implies a=ba = b or a=−ba = -b, we can write:
3x−1=10or3x−1=−103x - 1 = 10 \quad \text{or} \quad 3x - 1 = -10

STEP 4

First, solve the equation 3x−1=103x - 1 = 10.

STEP 5

Add 1 to both sides of the equation to isolate the term with the variable xx on one side.
3x−1+1=10+13x - 1 + 1 = 10 + 1

STEP 6

Simplify the equation.
3x=113x = 11

STEP 7

Divide both sides of the equation by 3 to solve for xx.
3x3=113\frac{3x}{3} = \frac{11}{3}

STEP 8

Simplify the equation to find the first solution for xx.
x=113x = \frac{11}{3}

STEP 9

Now, solve the second equation 3x−1=−103x - 1 = -10.

STEP 10

Add 1 to both sides of the equation to isolate the term with the variable xx on one side.
3x−1+1=−10+13x - 1 + 1 = -10 + 1

STEP 11

Simplify the equation.
3x=−93x = -9

STEP 12

Divide both sides of the equation by 3 to solve for xx.
3x3=−93\frac{3x}{3} = \frac{-9}{3}

STEP 13

Simplify the equation to find the second solution for xx.
x=−3x = -3
The solutions to the equation ∣3x−1∣=10|3x - 1| = 10 are x=113x = \frac{11}{3} and x=−3x = -3.

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