Let's solve each inequality step-by-step.
### Inequality 1: [tex]\( 4(9x - 18) > 3(8x + 12) \)[/tex]
1. Distribute the constants on both sides:
[tex]\[
4(9x - 18) \implies 4 \cdot 9x - 4 \cdot 18 = 36x - 72
\][/tex]
[tex]\[
3(8x + 12) \implies 3 \cdot 8x + 3 \cdot 12 = 24x + 36
\][/tex]
2. Rewrite the inequality with these new expressions:
[tex]\[
36x - 72 > 24x + 36
\][/tex]
3. Subtract [tex]\(24x\)[/tex] from both sides to isolate [tex]\(x\)[/tex]:
[tex]\[
36x - 24x - 72 > 24x - 24x + 36 \implies 12x - 72 > 36
\][/tex]
4. Add 72 to both sides to further isolate [tex]\(x\)[/tex]:
[tex]\[
12x - 72 + 72 > 36 + 72 \implies 12x > 108
\][/tex]
5. Divide both sides by 12:
[tex]\[
x > 9
\][/tex]
### Inequality 2: [tex]\(-\frac{1}{3}(12x + 6) \geq -2x + 14\)[/tex]
1. Distribute the [tex]\(-\frac{1}{3}\)[/tex] on the left side:
[tex]\[
-\frac{1}{3}(12x + 6) \implies -\frac{1}{3} \cdot 12x - \frac{1}{3} \cdot 6 = -4x - 2
\][/tex]
2. Rewrite the inequality with these new expressions:
[tex]\[
-4x - 2 \geq -2x + 14
\][/tex]
3. Add [tex]\(4x\)[/tex] to both sides to isolate [tex]\(x\)[/tex]:
[tex]\[
-4x + 4x - 2 \geq -2x + 4x + 14 \implies -2 \geq 2x + 14
\][/tex]
4. Subtract 14 from both sides to further isolate [tex]\(x\)[/tex]:
[tex]\[
-2 - 14 \geq 2x + 14 - 14 \implies -16 \geq 2x
\][/tex]
5. Divide both sides by 2:
[tex]\[
-8 \geq x \implies x \leq -8
\][/tex]
### Inequality 3: [tex]\(1.6(x + 8) \geq 38.4\)[/tex]
1. Distribute the 1.6 on the left side:
[tex]\[
1.6(x + 8) \implies 1.6 \cdot x + 1.6 \cdot 8 = 1.6x + 12.8
\][/tex]
2. Rewrite the inequality with these new expressions:
[tex]\[
1.6x + 12.8 \geq 38.4
\][/tex]
3. Subtract 12.8 from both sides to isolate [tex]\(x\)[/tex]:
[tex]\[
1.6x + 12.8 - 12.8 \geq 38.4 - 12.8 \implies 1.6x \geq 25.6
\][/tex]
4. Divide both sides by 1.6:
[tex]\[
x \geq 16
\][/tex]
In summary, the solutions to the inequalities are:
1. [tex]\( 4(9x - 18) > 3(8x + 12) \)[/tex] simplifies to [tex]\( x > 9 \)[/tex]
2. [tex]\( -\frac{1}{3}(12x + 6) \geq -2x + 14 \)[/tex] simplifies to [tex]\( x \leq -8 \)[/tex]
3. [tex]\( 1.6(x + 8) \geq 38.4 \)[/tex] simplifies to [tex]\( x \geq 16 \)[/tex]
