To solve the given inequalities: [tex]\( x - \frac{4}{3} x + 2 \leq 6 \)[/tex] and [tex]\( -2(x + 2) \geq 4 \)[/tex], let's go step-by-step.
### Inequality 1: [tex]\( x - \frac{4}{3} x + 2 \leq 6 \)[/tex]
1. Combine the terms involving [tex]\( x \)[/tex]:
[tex]\[
x - \frac{4}{3} x + 2 \leq 6
\][/tex]
[tex]\[
\left(1 - \frac{4}{3}\right) x + 2 \leq 6
\][/tex]
[tex]\[
-\frac{1}{3} x + 2 \leq 6
\][/tex]
2. Subtract 2 from both sides to isolate the term involving [tex]\( x \)[/tex]:
[tex]\[
-\frac{1}{3} x + 2 - 2 \leq 6 - 2
\][/tex]
[tex]\[
-\frac{1}{3} x \leq 4
\][/tex]
3. Multiply both sides by -3 to solve for [tex]\( x \)[/tex] (note that multiplying by a negative number reverses the inequality):
[tex]\[
x \geq -12
\][/tex]
Thus, the solution set for [tex]\( x - \frac{4}{3} x + 2 \leq 6 \)[/tex] is:
[tex]\[
x \geq -12
\][/tex]
### Inequality 2: [tex]\( -2(x + 2) \geq 4 \)[/tex]
1. Distribute the [tex]\(-2\)[/tex] on the left-hand side:
[tex]\[
-2(x + 2) \geq 4
\][/tex]
[tex]\[
-2x - 4 \geq 4
\][/tex]
2. Add 4 to both sides to isolate the term involving [tex]\( x \)[/tex]:
[tex]\[
-2x - 4 + 4 \geq 4 + 4
\][/tex]
[tex]\[
-2x \geq 8
\][/tex]
3. Divide both sides by -2 to solve for [tex]\( x \)[/tex] (note that dividing by a negative number reverses the inequality):
[tex]\[
x \leq -4
\][/tex]
Thus, the solution set for [tex]\( -2(x + 2) \geq 4 \)[/tex] is:
[tex]\[
x \leq -4
\][/tex]
### Combining the Results
The final step is to find the intersection of the two solution sets:
1. From the first inequality, we have [tex]\( x \geq -12 \)[/tex].
2. From the second inequality, we have [tex]\( x \leq -4 \)[/tex].
Combining these, the solution must satisfy both conditions:
[tex]\[
-12 \leq x \leq -4
\][/tex]
Therefore, the solution for the system of inequalities is:
[tex]\[
-12 \leq x \leq -4
\][/tex]
This means [tex]\( x \)[/tex] lies in the interval [tex]\([-12, -4]\)[/tex].
