To solve the inequality \(\left| 2x - 3 \right| \leq 11\), we can break it down into two separate inequalities:
- \(2x - 3 \leq 11\)
- \(2x - 3 \geq -11\)
Let's solve each inequality step by step.
1. Solve \(2x - 3 \leq 11\)
Add 3 to both sides:
\[ 2x \leq 11 + 3 \] \[ 2x \leq 14 \]
Now, divide both sides by 2:
\[ x \leq 7 \]
2. Solve \(2x - 3 \geq -11\)
Add 3 to both sides:
\[ 2x \geq -11 + 3 \] \[ 2x \geq -8 \]
Now, divide both sides by 2:
\[ x \geq -4 \]
Combining the Results
Now we combine the results from both inequalities:
\[ -4 \leq x \leq 7 \]
In interval notation, the solution is:
\[ [-4, 7] \]