Question

Solve the equation. Check the solutions.

$|3x-7|-5=-3$

Short answer

The values are $x=3,\frac{5}{3}$.

Step-by-step solution

Step 1- Apply the concept of absolute value.

Absolute value describes the distance from zero that a number is on the number line,without considering direction. The absolute value of a number is never negative.

Step 2- Simplify the expression.

Consider the expression $|3x-7|-5=-3$.

Add 5 on both sides of the equation as follows:

$\begin{array}{l}|3x-7|-5=-3\\ |3x-7|-5+5=-3+5\\ |3x-7|=2\end{array}$

Step 3- Step description.

Apply the absolute rule as follows:

Case 1. When $3x-7=-2$.

$\begin{array}{l}3x-7=-2\\ 3x-7+2=0\\ 3x-5=0\\ 3x=5\\ x=\frac{5}{3}\end{array}$

Case 2. When $3x-7=2$

$\begin{array}{l}3x-7=2\\ 3x-7-2=0\\ 3x-9=0\\ 3x=9\\ x=3\end{array}$

Therefore, the value is$x=3,\frac{5}{3}$.

Step 4-Verify the solutions.

Substitute $x=3,\frac{5}{3}$in the equation $|3x-7|-5=-3$and simplify as follows:

$\begin{array}{l}|3x-7|-5=-3\\ |3\left(3\right)-7|-5=-3\\ |9-7|-5=-3\\ \left|2\right|-5=-3\\ 2-5=-3\\ -3=-3\end{array}$

$\begin{array}{l}|3x-7|-5=-3\\ |3\left(\frac{5}{3}\right)-7|-5=-3\\ |5-7|-5=-3\\ |-2|-5=-3\\ 2-5=-3\\ -3=-3\end{array}$

Since the left-hand side and right-hand side is equal in both the cases thus the values are $x=3,\frac{5}{3}$.