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

First, start with \(-5(4x - 1)>-55\). Divide both sides by \(- 5\). Remember that when dividing an inequality by a negative number, the direction of the inequality sign changes. So we have \(4x - 1 < 11\).

$$4x-1<11$$

Step2: Isolate the variable \(x\)

Add \(1\) to both sides of the inequality \(4x - 1 < 11\): \(4x<12\). Then divide both sides by \(4\) to get \(x < 3\). In set - notation, the solution set is \(\{x|x < 3\}\).

Step3: Solve the second - inequality

For \(3\leqslant4x - 9\leqslant11\), add \(9\) to all parts of the compound inequality: \(3 + 9\leqslant4x-9 + 9\leqslant11 + 9\), which simplifies to \(12\leqslant4x\leqslant20\).

$$12\leqslant4x\leqslant20$$

Step4: Isolate the variable \(x\) in the second - inequality

Divide all parts of \(12\leqslant4x\leqslant20\) by \(4\): \(3\leqslant x\leqslant5\). In set - notation, the solution set is \(\{x|3\leqslant x\leqslant5\}\).

Step5: Graph the first solution set

On a number line for \(x < 3\), draw an open circle at \(x = 3\) (because \(3\) is not included in the solution set) and shade the line to the left of \(3\).

Step6: Graph the second solution set

On a number line for \(3\leqslant x\leqslant5\), draw a closed circle at \(x = 3\) and \(x = 5\) (because \(3\) and \(5\) are included in the solution set) and shade the line between \(3\) and \(5\).

Answer:

For \(-5(4x - 1)>-55\):
Solution Set: \(\{x|x < 3\}\)
Graph: Open - circle at \(3\), shade to the left.
For \(3\leqslant4x - 9\leqslant11\):
Solution Set: \(\{x|3\leqslant x\leqslant5\}\)
Graph: Closed - circles at \(3\) and \(5\), shade between \(3\) and \(5\).