Let's solve the inequalities step by step.
### Inequality 1: [tex]\(\frac{4x + 2}{5} \geq x + 1\)[/tex]
1. Remove the fraction by multiplying both sides by 5:
[tex]\[
4x + 2 \geq 5(x + 1)
\][/tex]
2. Distribute the 5 on the right-hand side:
[tex]\[
4x + 2 \geq 5x + 5
\][/tex]
3. Move all terms involving [tex]\(x\)[/tex] to one side and constants to the other:
[tex]\[
4x + 2 - 5x \geq 5 - 5x - 5x
\][/tex]
Simplify:
[tex]\[
-x + 2 \geq 5
\][/tex]
4. Isolate the variable [tex]\(x\)[/tex]:
[tex]\[
-x \geq 5 - 2
\][/tex]
[tex]\[
-x \geq 3
\][/tex]
5. Multiply both sides by -1 (and remember to reverse the inequality sign):
[tex]\[
x \leq -3
\][/tex]
So, the solution to the first inequality is:
[tex]\[ x \leq -3 \][/tex]
### Inequality 2: [tex]\(\frac{x}{2} - 1 > \frac{x}{3}\)[/tex]
1. Eliminate the fractions by finding a common denominator. In this case, use 6:
[tex]\[
3\left( \frac{x}{2} - 1 \right) > 2\left( \frac{x}{3} \right)
\][/tex]
2. Distribute the factors inside the parentheses:
[tex]\[
\frac{3x}{2} - 3 > \frac{2x}{3}
\][/tex]
3. Clear the denominators by multiplying every term by 6:
[tex]\[
6 \cdot \left( \frac{3x}{2} - 3 \right) > 6 \cdot \frac{2x}{3}
\][/tex]
Simplify:
[tex]\[
9x - 18 > 4x
\][/tex]
4. Move all terms involving [tex]\(x\)[/tex] to one side:
[tex]\[
9x - 4x > 18
\][/tex]
Simplify:
[tex]\[
5x > 18
\][/tex]
5. Isolate [tex]\(x\)[/tex]:
[tex]\[
x > \frac{18}{5}
\][/tex]
Simplify:
[tex]\[
x > 6
\][/tex]
So, the solution to the second inequality is:
[tex]\[ x > 6 \][/tex]
### Combining the Results
- The solution to the first inequality is [tex]\( x \leq -3 \)[/tex].
- The solution to the second inequality is [tex]\( x > 6 \)[/tex].
Thus, the solutions to the inequalities are:
[tex]\[ x \leq -3 \quad \text{or} \quad x > 6 \][/tex]
