Algebra 1 - Solving and Graphing Compound Inequalities Practice - Free Printable

Here are the step-by-step solutions for the compound inequality problems shown in your worksheet.

Part 1: Graph the compound inequality shown

1. $-3 < x \le 2$
* Logic: $x$ is greater than $-3$ (open circle) AND less than or equal to $2$ (closed circle). Shade between them.
* Graph: Open circle at $-3$, closed circle at $2$, shaded line connecting them.

2. $-5 < x \le -3$
* Logic: $x$ is greater than $-5$ (open circle) AND less than or equal to $-3$ (closed circle).
* Graph: Open circle at $-5$, closed circle at $-3$, shaded line connecting them.

3. $1 \le x \le 5$
* Logic: $x$ is between $1$ and $5$, inclusive.
* Graph: Closed circle at $1$, closed circle at $5$, shaded line connecting them.

4. $-1 \le x \le 3$
* Logic: $x$ is between $-1$ and $3$, inclusive.
* Graph: Closed circle at $-1$, closed circle at $3$, shaded line connecting them.

5. $-5 < x \le 0$
* Logic: $x$ is greater than $-5$ (open) AND less than or equal to $0$ (closed).
* Graph: Open circle at $-5$, closed circle at $0$, shaded line connecting them.

---

Part 2: Graph the compound inequality shown

6. $x \le -2$ or $x > 3$
* Logic: Two separate regions. One goes left from $-2$ (inclusive), one goes right from $3$ (exclusive).
* Graph: Closed circle at $-2$ with an arrow pointing left. Open circle at $3$ with an arrow pointing right.

7. $x < -3$ or $x \ge 1$
* Logic: Two separate regions. Left from $-3$ (exclusive), right from $1$ (inclusive).
* Graph: Open circle at $-3$ with an arrow pointing left. Closed circle at $1$ with an arrow pointing right.

8. $x \le -2$ or $x \ge 1$
* Logic: Two separate regions. Left from $-2$ (inclusive), right from $1$ (inclusive).
* Graph: Closed circle at $-2$ with an arrow pointing left. Closed circle at $1$ with an arrow pointing right.

9. $x \le -3$ or $x \ge 2$
* Logic: Two separate regions. Left from $-3$ (inclusive), right from $2$ (inclusive).
* Graph: Closed circle at $-3$ with an arrow pointing left. Closed circle at $2$ with an arrow pointing right.

10. $x < -3$ or $x \ge 4$
* Logic: Two separate regions. Left from $-3$ (exclusive), right from $4$ (inclusive).
* Graph: Open circle at $-3$ with an arrow pointing left. Closed circle at $4$ with an arrow pointing right.

---

Part 3: Solve the compound inequality shown

11. $-2 < 2x + 4$ and $-1 < 4 \le 2$
* *Note: The second part of this problem text seems to contain a typo or is nonsensical ($-1 < 4 \le 2$ is false because 4 is not $\le 2$). However, looking at the handwritten answer key, it seems the student ignored the second part or there was a specific context. Let's solve just the first valid inequality part:*
* $-2 < 2x + 4$
* Subtract 4 from both sides: $-6 < 2x$
* Divide by 2: $-3 < x$
* Answer: $x > -3$

12. $-5 \le 3x + 1 < -2$ OR $9 \le 2$
* *Note: $9 \le 2$ is false, so we ignore the "OR" part. We just solve the middle part.*
* Subtract 1 from all parts: $-6 \le 3x < -3$
* Divide all parts by 3: $-2 \le x < -1$
* Answer: $-2 \le x < -1$

13. $-6 < 4x + 9 < -2$ OR $-6 < x < 1$
* First, solve $-6 < 4x + 9 < -2$:
* Subtract 9: $-15 < 4x < -11$
* Divide by 4: $-3.75 < x < -2.75$
* Now look at the "OR": $-6 < x < 1$.
* Since the range $(-3.75, -2.75)$ is completely inside the range $(-6, 1)$, the union of these two sets is just the larger range.
* Answer: $-6 < x < 1$

14. $-2x \ge 5x + 7$ OR $1 \le 5 \le 4$
* *Note: $1 \le 5 \le 4$ is false. Solve the first part.*
* Subtract $5x$ from both sides: $-7x \ge 7$
* Divide by $-7$ (flip the sign): $x \le -1$
* Answer: $x \le -1$

15. $-5 < 2x - 1 < 4$ OR $x < 3$
* First, solve $-5 < 2x - 1 < 4$:
* Add 1: $-4 < 2x < 5$
* Divide by 2: $-2 < x < 2.5$
* Now look at the "OR": $x < 3$.
* The range $(-2, 2.5)$ is inside $x < 3$. So the combined answer is just $x < 3$.
* Answer: $x < 3$

---

Part 4: Solve the compound inequality shown

16. $2x + 3 < -3$ OR $x - 2 > 1$
* Left side: $2x < -6 \rightarrow x < -3$
* Right side: $x > 3$
* Answer: $x < -3$ or $x > 3$

17. $5x < -15$ OR $3x > 15$
* Left side: Divide by 5 $\rightarrow x < -3$
* Right side: Divide by 3 $\rightarrow x > 5$
* Answer: $x < -3$ or $x > 5$

18. $7x - 3 \ge 3x + 5$ OR $2x + 9 \ge 19$
* Left side:
* Subtract $3x$: $4x - 3 \ge 5$
* Add 3: $4x \ge 8$
* Divide by 4: $x \ge 2$
* Right side:
* Subtract 9: $2x \ge 10$
* Divide by 2: $x \ge 5$
* Combine with OR: If $x \ge 2$, it covers everything in $x \ge 5$. The wider condition wins.
* Answer: $x \ge 2$

19. $-5x + 2 > -3(x + 2)$ OR $-13 \ge 4x + 7$
* Left side:
* Distribute: $-5x + 2 > -3x - 6$
* Add $5x$: $2 > 2x - 6$
* Add 6: $8 > 2x$
* Divide by 2: $4 > x$ (or $x < 4$)
* Right side:
* Subtract 7: $-20 \ge 4x$
* Divide by 4: $-5 \ge x$ (or $x \le -5$)
* Combine with OR: $x < 4$ includes all numbers less than 4. $x \le -5$ is already included in that set.
* Answer: $x < 4$

20. $-4x - 3(x + 5) \ge 6$ OR $x + 5 > 3$
* Left side:
* Distribute: $-4x - 3x - 15 \ge 6$
* Combine: $-7x - 15 \ge 6$
* Add 15: $-7x \ge 21$
* Divide by -7 (flip sign): $x \le -3$
* Right side:
* Subtract 5: $x > -2$
* Combine with OR: These are two separate pieces. One piece is numbers less than or equal to -3. The other is numbers strictly greater than -2. They do not overlap, and they don't cover the gap between -3 and -2.
* Answer: $x \le -3$ or $x > -2$

Final Answer:
Part 1
1. Graph: Open circle at -3, closed circle at 2, shade between.
2. Graph: Open circle at -5, closed circle at -3, shade between.
3. Graph: Closed circle at 1, closed circle at 5, shade between.
4. Graph: Closed circle at -1, closed circle at 3, shade between.
5. Graph: Open circle at -5, closed circle at 0, shade between.

Part 2
6. Graph: Arrow left from closed circle at -2; Arrow right from open circle at 3.
7. Graph: Arrow left from open circle at -3; Arrow right from closed circle at 1.
8. Graph: Arrow left from closed circle at -2; Arrow right from closed circle at 1.
9. Graph: Arrow left from closed circle at -3; Arrow right from closed circle at 2.
10. Graph: Arrow left from open circle at -3; Arrow right from closed circle at 4.

Part 3
11. $x > -3$
12. $-2 \le x < -1$
13. $-6 < x < 1$
14. $x \le -1$
15. $x < 3$

Part 4
16. $x < -3$ or $x > 3$
17. $x < -3$ or $x > 5$
18. $x \ge 2$
19. $x < 4$
20. $x \le -3$ or $x > -2$