Question

1. $-3 < 6m + 10 < 34$
2. $-5 + 8r < -17$ or $9r + 8 > 17$

solve each inequality and graph its solution.

1. $-22 \leq 2 - 3m \leq 14$
2. $-5 - r > 7$

solve for the indicated variable in parenthesis.

1. $\frac{a + b}{k} = 1$ \\ \\ $(b)$
2. $4x - 5y = 9$ \\ \\ $(y)$
3. $4x - 5y = 9$ \\ \\ $(y)$
4. find the domain of the following graph.

unit 2

Explanation:

Problem 11: Solve \(-22 \leq 2 - 3m \leq 14\) and graph the solution.

Step 1: Subtract 2 from all parts

To isolate the term with \(m\), we subtract 2 from each part of the compound inequality.
\(-22 - 2 \leq 2 - 3m - 2 \leq 14 - 2\)
Simplifying each part:
\(-24 \leq -3m \leq 12\)

Step 2: Divide by -3 (and reverse inequalities)

When dividing by a negative number, the direction of the inequality signs reverses.
Divide each part by \(-3\):
\(\frac{-24}{-3} \geq \frac{-3m}{-3} \geq \frac{12}{-3}\)
Simplifying:
\(8 \geq m \geq -4\)
Which can be rewritten as \(-4 \leq m \leq 8\).

Problem 12: Solve \(-5 - r > 7\) and graph the solution.

Step 1: Add 5 to both sides

To isolate the term with \(r\), add 5 to both sides of the inequality.
\(-5 - r + 5 > 7 + 5\)
Simplifying:
\(-r > 12\)

Step 2: Multiply by -1 (and reverse inequality)

Multiply both sides by \(-1\) (remember to reverse the inequality sign).
\(r < -12\)

Problem 13: Solve \(\frac{a + b}{k} = 1\) for \(b\).

Step 1: Multiply both sides by \(k\)

To eliminate the denominator, multiply both sides by \(k\).
\(k \cdot \frac{a + b}{k} = 1 \cdot k\)
Simplifying:
\(a + b = k\)

Step 2: Subtract \(a\) from both sides

Subtract \(a\) from both sides to solve for \(b\).
\(a + b - a = k - a\)
Simplifying:
\(b = k - a\)

Problem 14: Solve \(4x - 5y = 9\) for \(y\).

Answer:

\( -2 < m < 4 \)

1. \( -5 + 8r < -17 \) or \( 9r + 8 > 17 \)

First inequality: \( 8r < -12 \) → \( r < -\frac{3}{2} \)
Second inequality: \( 9r > 9 \) → \( r > 1 \)