Question

1. |7x - 2| = x + 4

show your work below your answer. draw the appropriate arrows or points or drag the provided arrows.
(number line from -10 to 10 is shown)
answer:

Explanation:

Step1: Consider two cases for absolute value

Case 1: \(7x - 2 = x + 4\)

Step2: Solve Case 1 equation

Subtract \(x\) from both sides: \(7x - x - 2 = 4\)
Simplify: \(6x - 2 = 4\)
Add 2 to both sides: \(6x = 4 + 2\)
\(6x = 6\)
Divide by 6: \(x = 1\)

Case 2: \(7x - 2 = -(x + 4)\)

Step3: Solve Case 2 equation

Expand the right side: \(7x - 2 = -x - 4\)
Add \(x\) to both sides: \(7x + x - 2 = -4\)
Simplify: \(8x - 2 = -4\)
Add 2 to both sides: \(8x = -4 + 2\)
\(8x = -2\)
Divide by 8: \(x = -\frac{2}{8} = -\frac{1}{4}\)

Step4: Check solutions

For \(x = 1\): Left side \(|7(1) - 2| = |5| = 5\), Right side \(1 + 4 = 5\), valid.
For \(x = -\frac{1}{4}\): Left side \(|7(-\frac{1}{4}) - 2| = |-\frac{7}{4} - \frac{8}{4}| = |-\frac{15}{4}| = \frac{15}{4}\), Right side \(-\frac{1}{4} + 4 = \frac{15}{4}\), valid.

Answer:

\(x = 1\) or \(x = -\frac{1}{4}\)