Question

Solve the equation and check the solution.

$|4x-5|=-25$

Short answer

No solution.

Step-by-step solution

Step 1- Apply the concept of absolute value.

For any real numbers $a,b$, where $b\ge 0$, if $\left|a\right|=b$ then $a=b$ or $a=-b$.

Step 2- Step description.

Consider the equation $|4x-5|=-25$.

Use the concept of absolute value equation as follows:

$4x-5=-25$ or $4x-5=25$.

Step 3- Step description.

Case 1. Simplify $4x-5=25$.

Subtract 25 from both sides of the equation and simplify as follows:

$\begin{array}{l}4x-5=25\\ 4x-5-25=25-25\\ 4x-30=0\\ 4x=30\\ x=\frac{30}{4}\end{array}$

Case 2. Simplify $4x-5=-25$.

Add 25 on both sides of the equation and simplify as follows:

$\begin{array}{l}4x-5=-25\\ 4x-5+25=-25+25\\ 4x+20=0\\ 4x=-20\\ x=\frac{-20}{4}\\ x=-5\end{array}$

As the solution is negative, but the absolute value cannot be negative, therefore no solution for $x\in \mathrm{ℝ}$.