Question

1. solve the inequality, write the solution in set notation, and graph the solution set. -5(4x - 1) > -55

solution set:
graph:

1. solve the inequality, write the solution in set notation, and graph the solution set. 3 ≤ 4x - 9 ≤ 11

solution set:
graph:

Explanation:

Step1: Solve the first - inequality

Divide both sides of $-5(4x - 1)>-55$ by $- 5$. Remember to reverse the inequality sign when dividing by a negative number.
\[

$$\begin{align*} \frac{-5(4x - 1)}{-5}&<\frac{-55}{-5}\\ 4x-1&<11 \end{align*}$$

\]

Step2: Isolate the variable $x$

Add 1 to both sides of the inequality $4x - 1<11$.
\[

$$\begin{align*} 4x-1 + 1&<11+1\\ 4x&<12 \end{align*}$$

\]

Step3: Solve for $x$

Divide both sides of $4x<12$ by 4.
\[

$$\begin{align*} \frac{4x}{4}&<\frac{12}{4}\\ x&<3 \end{align*}$$

\]
The solution set in set - notation is $\{x|x < 3\}$.

Step4: Solve the second - inequality

For the compound inequality $3\leq4x - 9\leq11$, first add 9 to all parts of the inequality.
\[

$$\begin{align*} 3+9&\leq4x-9 + 9\leq11 + 9\\ 12&\leq4x\leq20 \end{align*}$$

\]

Step5: Solve for $x$ in the compound inequality

Divide all parts of the inequality $12\leq4x\leq20$ by 4.
\[

$$\begin{align*} \frac{12}{4}&\leq\frac{4x}{4}\leq\frac{20}{4}\\ 3&\leq x\leq5 \end{align*}$$

\]
The solution set in set - notation is $\{x|3\leq x\leq5\}$.

Answer:

For $-5(4x - 1)>-55$:
Solution Set: $\{x|x < 3\}$
Graph: An open - circle at 3 on the number line and a line extending to the left.

For $3\leq4x - 9\leq11$:
Solution Set: $\{x|3\leq x\leq5\}$
Graph: A closed - circle at 3, a closed - circle at 5, and a line segment connecting them.