Question

1. (2leq -3(2x - 6)leq -9x + 4)

Explanation:

Step1: Expand the middle - part

First, expand $-3(2x - 6)$ using the distributive property $a(b - c)=ab - ac$. Here, $a=-3$, $b = 2x$, and $c = 6$. So, $-3(2x - 6)=-6x+18$. The inequality becomes $2\leq - 6x + 18\leq - 9x + 4$.

Step2: Split into two inequalities

Split the compound - inequality $2\leq - 6x + 18\leq - 9x + 4$ into two inequalities:
Inequality 1: $2\leq - 6x + 18$. Subtract 18 from both sides: $2-18\leq - 6x+18 - 18$, which simplifies to $-16\leq - 6x$. Divide both sides by $-6$. When dividing an inequality by a negative number, the direction of the inequality sign changes. So, $\frac{-16}{-6}\geq x$, or $\frac{8}{3}\geq x$.
Inequality 2: $-6x + 18\leq - 9x + 4$. Add $9x$ to both sides: $-6x+9x + 18\leq - 9x+9x + 4$, which gives $3x + 18\leq 4$. Subtract 18 from both sides: $3x+18 - 18\leq 4 - 18$, so $3x\leq - 14$. Divide both sides by 3: $x\leq-\frac{14}{3}$.

Step3: Find the intersection

We need to find the values of $x$ that satisfy both inequalities. Since $\frac{8}{3}\geq x$ and $x\leq-\frac{14}{3}$, and $-\frac{14}{3}\approx - 4.67$ and $\frac{8}{3}\approx2.67$, the intersection of the two solution - sets is $x\leq-\frac{14}{3}$.

Answer:

$x\leq-\frac{14}{3}$