Question

solve and graph each. show your work.

1. $-1 < 9 + n < 17$

(number line: -12, -10, -8, -6, -4, -2, 0, 2, 4, 6, 8)

1. $-3 \leq \frac{p}{2} < 0$

(number line: -10, -8, -6, -4, -2, 0, 2)
determine if it is an and inequality or an or inequality. write the inequality.
(number line: -6, -3, 0, 3, 6, 9, 12 with a closed dot at -3 and open dot at 9)
(number line: -3, -1, 1, 3, 5, 7, 9 with open dot at -1 and closed dot at 5)

Explanation:

Problem 2: Solve \(-1 < 9 + n < 17\)

Step 1: Subtract 9 from all parts

To isolate \(n\), we subtract 9 from each part of the compound inequality.
\[
-1 - 9 < 9 + n - 9 < 17 - 9
\]

Step 2: Simplify each part

Simplify the left, middle, and right expressions.
\[
-10 < n < 8
\]

Step 1: Multiply all parts by 2

To isolate \(p\), we multiply each part of the compound inequality by 2.
\[
-3 \times 2 \leq \frac{p}{2} \times 2 < 0 \times 2
\]

Step 2: Simplify each part

Simplify the left, middle, and right expressions.
\[
-6 \leq p < 0
\]

A closed circle at \(-3\) means \(\geq -3\) (or \(\leq -3\), but direction here is right from \(-3\)) and an open circle at \(9\) means \(< 9\). The shaded region is between \(-3\) and \(9\), so it's an "and" inequality (both conditions must hold: \(x \geq -3\) and \(x < 9\)).

Answer:

\(-10 < n < 8\) (For graphing, draw an open circle at \(-10\) and \(8\) on the number line and shade the region between them.)

Problem 5: Solve \(-3 \leq \frac{p}{2} < 0\)