Sure! Let's solve the inequality step by step.
1. Start with the given inequality:
[tex]\[
|x - 12| + 5 < 27
\][/tex]
2. Subtract 5 from both sides to isolate the absolute value term:
[tex]\[
|x - 12| < 22
\][/tex]
3. To remove the absolute value, we need to create two separate inequalities:
[tex]\[
x - 12 < 22
\][/tex]
and
[tex]\[
-(x - 12) < 22
\][/tex]
4. Solve the first inequality:
[tex]\[
x - 12 < 22 \implies x < 34
\][/tex]
5. Solve the second inequality. Start by distributing the negative sign:
[tex]\[
-(x - 12) < 22 \implies x - 12 > -22
\][/tex]
Then add 12 to both sides:
[tex]\[
x > -10
\][/tex]
So, the two equations created by the inequality are:
[tex]\[
y_1 = x < 34
\][/tex]
[tex]\[
y_2 = x > -10
\][/tex]
Hence, the inequalities are:
[tex]\[
x < 34 \quad \text{and} \quad x > -10
\][/tex]
