To solve the inequality [tex]\( 2 - \left| \frac{2}{5} x + 3 \right| > \frac{3}{5} \)[/tex], follow these detailed steps:
1. Isolate the Absolute Value Expression:
[tex]\[
2 - \left| \frac{2}{5} x + 3 \right| > \frac{3}{5}
\][/tex]
Subtract [tex]\( \frac{3}{5} \)[/tex] from both sides:
[tex]\[
2 - \frac{3}{5} > \left| \frac{2}{5} x + 3 \right|
\][/tex]
2. Simplify the Left Side:
[tex]\[
\frac{10}{5} - \frac{3}{5} = \frac{7}{5}
\][/tex]
So the inequality is:
[tex]\[
\frac{7}{5} > \left| \frac{2}{5} x + 3 \right|
\][/tex]
3. Interpret the Absolute Value Inequality:
[tex]\[
\left| \frac{2}{5} x + 3 \right| < \frac{7}{5}
\][/tex]
This absolute value inequality can be split into two separate inequalities:
[tex]\[
-\frac{7}{5} < \frac{2}{5} x + 3 < \frac{7}{5}
\][/tex]
4. Solve Each Inequality:
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]
Convert 3 into a fraction with the same denominator:
[tex]\[
-\frac{7}{5} - \frac{15}{5} = -\frac{22}{5}
\][/tex]
So the inequality becomes:
[tex]\[
-\frac{22}{5} < \frac{2}{5} x
\][/tex]
Multiply both sides by [tex]\( \frac{5}{2} \)[/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]
Convert 3 into a fraction with the same denominator:
[tex]\[
3 = \frac{15}{5}
\][/tex]
So the inequality becomes:
[tex]\[
\frac{2}{5} x < \frac{7}{5} - \frac{15}{5} = -\frac{8}{5}
\][/tex]
Multiply both sides by [tex]\( \frac{5}{2} \)[/tex]:
[tex]\[
x < -4
\][/tex]
5. Combine the Results:
Together, the inequalities [tex]\( x > -11 \)[/tex] and [tex]\( x < -4 \)[/tex] give us the solution set:
[tex]\[
-11 < x < -4
\][/tex]
Thus, the solution to the inequality [tex]\( 2 - \left| \frac{2}{5} x + 3 \right| > \frac{3}{5} \)[/tex] is:
[tex]\[
(-11, -4)
\][/tex]
