Question

solve the compound inequality.

(4x + 1 > -7) or (2x - 3 < -15)

graph the solution on the number line.

Answer:

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<reasoning>

Solve the first inequality

Using the Compound Inequalities knowledge point
\[

$$\begin{aligned} 4x + 1 &> -7 \\ 4x &> -8 \\ x &> -2 \end{aligned}$$

\]

Solve the second inequality

Using the Compound Inequalities knowledge point
\[

$$\begin{aligned} 2x - 3 &< -15 \\ 2x &< -12 \\ x &< -6 \end{aligned}$$

\]

Combine the solutions

Using the Compound Inequalities knowledge point
\[
x < -6 \quad \text{or} \quad x > -2
\]
In interval notation, this is written as:
\[
(-\infty, -6) \cup (-2, \infty)
\]

Graph the solution on a number line

Using the Inequality Graphing knowledge point

• Plot an open circle at \(x = -6\) and shade to the left.
• Plot an open circle at \(x = -2\) and shade to the right.

</reasoning>

<answer>
The solution to the compound inequality is:
\[
x < -6 \quad \text{or} \quad x > -2
\]
In interval notation:
\[
(-\infty, -6) \cup (-2, \infty)
\]
</answer>

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