Question

directions: solve each of the following inequalities, write the solution set two ways, graph the solution set, and select all of the solutions from the box.

1. $-5 > h - 4$
2. $-3n \geq 9$
3. $15 + x \leq 11$

Explanation:

Problem 1: Solve \(-5 > h - 4\)

Step 1: Add 4 to both sides

To isolate \(h\), we add 4 to both sides of the inequality.
\[
-5 + 4 > h - 4 + 4
\]

Step 2: Simplify both sides

Simplifying the left and right sides gives:
\[
-1 > h
\]
or \(h < -1\)

Now, we select the values from the box that are less than \(-1\). The values are: \(-7\), \(-11\), \(-8\), \(-5\), \(-63\), \(-6\), \(-9\), \(-4\) (wait, \(-4 < -1\), yes), \(-3\) (wait, \(-3 < -1\), yes), \(-2\) (wait, \(-2 < -1\), yes). Wait, let's list all values less than \(-1\):

Looking at the box:

• First row: \(-7\) (yes), \(3\) (no), \(-11\) (yes), \(0\) (no), \(1\) (no), \(20\) (no)
• Second row: \(-4\) (yes), \(2\) (no), \(-8\) (yes), \(17\) (no), \(-1\) (no, since \(h < -1\), not \(\leq\)), \(6\) (no)
• Third row: \(-2\) (yes), \(9\) (no), \(100\) (no), \(-5\) (yes), \(-3\) (yes), \(4\) (no)
• Fourth row: \(5\) (no), \(-63\) (yes), \(-6\) (yes), \(7\) (no), \(15\) (no), \(-9\) (yes)

So the solutions are: \(-7\), \(-11\), \(-4\), \(-8\), \(-2\), \(-5\), \(-3\), \(-63\), \(-6\), \(-9\)

Step 1: Divide both sides by -3 (reverse inequality)

When dividing by a negative number, the inequality sign reverses.
\[
\frac{-3n}{-3} \leq \frac{9}{-3}
\]

Step 2: Simplify both sides

Simplifying gives:
\[
n \leq -3
\]

Now, we select the values from the box that are less than or equal to \(-3\). The values are:

Looking at the box:

• First row: \(-7\) (yes), \(3\) (no), \(-11\) (yes), \(0\) (no), \(1\) (no), \(20\) (no)
• Second row: \(-4\) (yes), \(2\) (no), \(-8\) (yes), \(17\) (no), \(-1\) (no), \(6\) (no)
• Third row: \(-2\) (no, since \(-2 > -3\)), \(9\) (no), \(100\) (no), \(-5\) (yes), \(-3\) (yes), \(4\) (no)
• Fourth row: \(5\) (no), \(-63\) (yes), \(-6\) (yes), \(7\) (no), \(15\) (no), \(-9\) (yes)

So the solutions are: \(-7\), \(-11\), \(-4\), \(-8\), \(-5\), \(-3\), \(-63\), \(-6\), \(-9\)

Step 1: Subtract 15 from both sides

To isolate \(x\), we subtract 15 from both sides.
\[
15 + x - 15 \leq 11 - 15
\]

Step 2: Simplify both sides

Simplifying gives:
\[
x \leq -4
\]

Now, we select the values from the box that are less than or equal to \(-4\). The values are:

Looking at the box:

• First row: \(-7\) (yes), \(3\) (no), \(-11\) (yes), \(0\) (no), \(1\) (no), \(20\) (no)
• Second row: \(-4\) (yes), \(2\) (no), \(-8\) (yes), \(17\) (no), \(-1\) (no), \(6\) (no)
• Third row: \(-2\) (no), \(9\) (no), \(100\) (no), \(-5\) (yes), \(-3\) (no), \(4\) (no)
• Fourth row: \(5\) (no), \(-63\) (yes), \(-6\) (yes), \(7\) (no), \(15\) (no), \(-9\) (yes)

So the solutions are: \(-7\), \(-11\), \(-4\), \(-8\), \(-5\), \(-63\), \(-6\), \(-9\)

Answer:

Solution to \(-5 > h - 4\) is \(h < -1\). Solutions from the box: \(-7\), \(-11\), \(-4\), \(-8\), \(-2\), \(-5\), \(-3\), \(-63\), \(-6\), \(-9\)

Problem 2: Solve \(-3n \geq 9\)