Question

① write the sentence as an inequality graph the inequality a number p is greater than - 3 and less than 9
② solve 22 < 4x - 6 ≤ 38 graph the solution
③ solve x + 5 ≤ - 11 or 3x - 1 ≥ - 7 write the solution in set - builder notation graph the solution

Explanation:

Step1: Translate the first - sentence to an inequality

The statement "A number $p$ is greater than - 3 and less than 9" can be written as $-3 < p<9$. To graph it, draw a number line, mark - 3 and 9 with open circles (since the inequality is strict), and shade the region between them.

Step2: Solve the second inequality

Given $22 < 4x - 6\leq38$. First, add 6 to all parts of the compound - inequality: $22+6 < 4x-6 + 6\leq38 + 6$, which simplifies to $28 < 4x\leq44$. Then divide all parts by 4: $\frac{28}{4}<\frac{4x}{4}\leq\frac{44}{4}$, so $7 < x\leq11$. To graph it, mark 7 with an open circle and 11 with a closed circle on the number line and shade the region between them.

Step3: Solve the third compound - inequality

For $x + 10\leq - 11$ or $3x-1\geq - 7$.
For $x + 10\leq - 11$, subtract 10 from both sides: $x+10 - 10\leq - 11 - 10$, so $x\leq - 21$.
For $3x-1\geq - 7$, add 1 to both sides: $3x-1 + 1\geq - 7 + 1$, which gives $3x\geq - 6$. Then divide both sides by 3: $x\geq - 2$.
In set - builder notation, the solution is $\{x|x\leq - 21\text{ or }x\geq - 2\}$. To graph it, mark - 21 with a closed circle and shade to the left, mark - 2 with a closed circle and shade to the right.

Answer:

1. Inequality: $-3 < p<9$; Graph: Open circles at - 3 and 9, shade between.
2. Solution: $7 < x\leq11$; Graph: Open circle at 7, closed circle at 11, shade between.
3. Solution in set - builder notation: $\{x|x\leq - 21\text{ or }x\geq - 2\}$; Graph: Closed circle at - 21, shade left; closed circle at - 2, shade right.