Let's solve the given inequality step by step:
The inequality to solve is:
[tex]\[ 2 - \left| \frac{2}{5} x + 3 \right| > \frac{3}{5} \][/tex]
1. Isolate the absolute value expression:
First, subtract [tex]\(\frac{3}{5}\)[/tex] from both sides:
[tex]\[ 2 - \left| \frac{2}{5} x + 3 \right| > \frac{3}{5} \][/tex]
[tex]\[ 2 - \left| \frac{2}{5} x + 3 \right| - \frac{3}{5} > 0 \][/tex]
[tex]\[ 2 - \frac{3}{5} - \left| \frac{2}{5} x + 3 \right| > 0 \][/tex]
2. Simplify the left-hand side:
Combine the constants:
[tex]\[ 2 - \frac{3}{5} = \frac{10}{5} - \frac{3}{5} = \frac{7}{5} \][/tex]
So, our inequality now looks like:
[tex]\[ \frac{7}{5} - \left| \frac{2}{5} x + 3 \right| > 0 \][/tex]
3. Isolate the absolute value term:
Rearrange to get:
[tex]\[ -\left| \frac{2}{5} x + 3 \right| > -\frac{7}{5} \][/tex]
Since multiplying or dividing an inequality by a negative number reverses the inequality sign, we get:
[tex]\[ \left| \frac{2}{5} x + 3 \right| < \frac{7}{5} \][/tex]
4. Solve the inequality without the absolute value:
The inequality [tex]\( \left| \frac{2}{5} x + 3 \right| < \frac{7}{5} \)[/tex] can be split into two inequalities:
[tex]\[ -\frac{7}{5} < \frac{2}{5} x + 3 < \frac{7}{5} \][/tex]
5. Solve for [tex]\(x\)[/tex] in both inequalities:
First inequality:
[tex]\[ -\frac{7}{5} < \frac{2}{5} x + 3 \][/tex]
Subtract 3 from both sides:
[tex]\[ -\frac{7}{5} - 3 < \frac{2}{5} x \][/tex]
[tex]\[ -\frac{7}{5} - \frac{15}{5} < \frac{2}{5} x \][/tex]
[tex]\[ -\frac{22}{5} < \frac{2}{5} x \][/tex]
Multiply both sides by [tex]\(\frac{5}{2}\)[/tex] to isolate [tex]\(x\)[/tex]:
[tex]\[ x > -11 \][/tex]
Second inequality:
[tex]\[ \frac{2}{5} x + 3 < \frac{7}{5} \][/tex]
Subtract 3 from both sides:
[tex]\[ \frac{2}{5} x < \frac{7}{5} - 3 \][/tex]
[tex]\[ \frac{2}{5} x < \frac{7}{5} - \frac{15}{5} \][/tex]
[tex]\[ \frac{2}{5} x < -\frac{8}{5} \][/tex]
Multiply both sides by [tex]\(\frac{5}{2}\)[/tex] to isolate [tex]\(x\)[/tex]:
[tex]\[ x < -4 \][/tex]
6. Combine the solutions:
Combining both parts, we get:
[tex]\[ -11 < x < -4 \][/tex]
Thus, the solution to the inequality is:
[tex]\[ -11 < x < -4 \][/tex]
