Let's tackle the given intervals step by step.

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10. Interval: $(-\infty, 2]$

Part A: Write as an Inequality

• The interval $(-\infty, 2]$ means all values less than or equal to 2.
• In inequality form:
  $x \leq 2$

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Part B: Graph the Solution

• How to graph $x \leq 2$:
  • Draw a number line.
  • Mark $2$ on the line and use a filled circle to show that $2$ is included (due to the $\leq$ sign).
  • Shade everything to the left of $2$ to indicate all values less than or equal to $2$.

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11. Interval: $(-\infty, -1]$ or $(3, \infty)$

Part A: Write as an Inequality

• The first interval $(-\infty, -1]$ means all values less than or equal to $-1$.
  In inequality form:
  $x \leq -1$

• The second interval $(3, \infty)$ means all values greater than $3$.
  In inequality form:
  $x > 3$

• Combine these two using "or":
  $x \leq -1 \quad \text{or} \quad x > 3$

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Part B: Graph the Solution

• How to graph $x \leq -1$ or $x > 3$:
  • Draw a number line.
  • For $x \leq -1$:
    • Mark $-1$ with a filled circle (since $-1$ is included) and shade everything to the left of $-1$.
  • For $x > 3$:
    • Mark $3$ with an open circle (since $3$ is not included) and shade everything to the right of $3$.

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Would You Like Graphs?

Let me know if you’d like me to create visual graphs for these solutions!

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Questions for Further Practice:

1. How would you express the interval $(4, \infty)$ as an inequality?
2. Rewrite $[-5, 7)$ as an inequality and graph it.
3. Can $(-\infty, \infty)$ be written as an inequality? Why or why not?
4. What does the interval $(2, 2]$ represent? Explain its validity.
5. How would you graph the solution for $-2 \leq x < 4$?

Tip:

When interpreting intervals, remember that $[$ or $]$ includes the boundary point, while $($ or $)$ excludes it!