Question

algebra 1
code 18: solving and graphing comp. ineq. (case: or)
name
id: 1
date
period
solve each compound inequality and graph its solution.

1. $-4x + 1 \leq 1 - 3x$ or $5x - 7 \geq 6x + 8$
2. $-4r - 7 > -4 - 7r$ or $7r + 8 \leq -8 + 5r$
3. $3b - 4 > 5b + 2$ or $-4 + b \geq -4 - 6b$
4. $8 - 5m > -8m - 1$ or $3m - 10 > 5 + 4m$
5. $6 + 4v < -8v + 6$ or $-4v + 10 \leq -3v + 2$
6. $-3p + 1 > 10 - 4p$ or $3p + 3 > 8p - 2$
7. $9 + 5x \leq 6x + 2$ or $10x + 7 < 7 + x$
8. $6b + 2 \geq 3b - 1$ or $-3b - 3 > 7 - b$
9. $-5b - 6 \geq -4b + 8$ or $6 - 8b \geq -2 - 10b$
10. $8v - 10 \leq -v - 10$ or $8v - 5 \geq 4 + 7v$
11. $2 + 5n < 9n - 10$ or $-3n - 8 > 10n - 8$
12. $7r - 1 \leq 8r - 6$ or $r + 4 \geq 6r + 9$
13. $6 + 7n > 5n - 6$ or $6n - 5 > 9 + 8n$
14. $7n + 6 > 6 - n$ or $-7 + 9n > 10n - 1$
15. $-2 - n \geq -5n + 10$ or $2 - 3n < -4n + 4$
16. $5x + 2 > 9x + 2$ or $8x + 7 \leq 2 + 9x$
17. $3n + 10 < 5 + 2n$ or $5 + 8n \geq -1 + 6n$
18. $1 - 4x < -3x - 3$ or $x + 8 > 8x - 6$
19. $5m + 7 < 6m - 8$ or $-2 + 2m \geq 3m - 10$
20. $5 + 9n \geq 5 - 7n$ or $7n + 1 \leq 6n - 6$

Explanation:

1) Step1: Solve $-4x+1 \leq 1-3x$

Subtract 1, add $4x$:
$0 \leq x$ or $x \geq 0$

1) Step2: Solve $5x-7 \geq 6x+8$

Subtract $5x$, subtract 8:
$-15 \geq x$ or $x \leq -15$

1) Step3: Combine OR solutions

Solution: $x \leq -15$ or $x \geq 0$
---

2) Step1: Solve $-4r-7 > -4-7r$

Add $7r$, add 7:
$3r > 3 \implies r > 1$

2) Step2: Solve $7r+8 \leq -8+5r$

Subtract $5r$, subtract 8:
$2r \leq -16 \implies r \leq -8$

2) Step3: Combine OR solutions

Solution: $r \leq -8$ or $r > 1$
---

3) Step1: Solve $3b-4 > 5b+2$

Subtract $3b$, subtract 2:
$-6 > 2b \implies b < -3$

3) Step2: Solve $-4+b \geq -4-6b$

Add 4, add $6b$:
$7b \geq 0 \implies b \geq 0$

3) Step3: Combine OR solutions

Solution: $b < -3$ or $b \geq 0$
---

4) Step1: Solve $8-5m > -8m-1$

Add $8m$, subtract 8:
$3m > -9 \implies m > -3$

4) Step2: Solve $3m-10 > 5+4m$

Subtract $3m$, subtract 5:
$-15 > m$ or $m < -15$

4) Step3: Combine OR solutions

Solution: $m < -15$ or $m > -3$
---

5) Step1: Solve $6+4v < -8v+6$

Subtract 6, add $8v$:
$12v < 0 \implies v < 0$

5) Step2: Solve $-4v+10 \leq -3v+2$

Subtract 10, add $4v$:
$0 \leq v - 8 \implies v \geq 8$

5) Step3: Combine OR solutions

Solution: $v < 0$ or $v \geq 8$
---

6) Step1: Solve $-3p+1 > 10-4p$

Subtract 1, add $4p$:
$p > 9$

6) Step2: Solve $3p+3 > 8p-2$

Subtract $3p$, add 2:
$5 > 5p \implies p < 1$

6) Step3: Combine OR solutions

Solution: $p < 1$ or $p > 9$
---

7) Step1: Solve $9+5x \leq 6x+2$

Subtract 9, subtract $5x$:
$0 \leq x - 7 \implies x \geq 7$

7) Step2: Solve $10x+7 < 7+x$

Subtract 7, subtract $x$:
$9x < 0 \implies x < 0$

7) Step3: Combine OR solutions

Solution: $x < 0$ or $x \geq 7$
---

8) Step1: Solve $6b+2 \geq 3b-1$

Subtract 2, subtract $3b$:
$3b \geq -3 \implies b \geq -1$

8) Step2: Solve $-3b-3 > 7-b$

Add 3, add $b$:
$-2b > 10 \implies b < -5$

8) Step3: Combine OR solutions

Solution: $b < -5$ or $b \geq -1$
---

9) Step1: Solve $-5b-6 \geq -4b+8$

Add 6, add $5b$:
$0 \geq b + 14 \implies b \leq -14$

9) Step2: Solve $6-8b \geq -2-10b$

Subtract 6, add $10b$:
$2b \geq -8 \implies b \geq -4$

9) Step3: Combine OR solutions

Solution: $b \leq -14$ or $b \geq -4$
---

10) Step1: Solve $8v-10 \leq -v-10$

Add 10, add $v$:
$9v \leq 0 \implies v \leq 0$

10) Step2: Solve $8v-5 \geq 4+7v$

Add 5, subtract $7v$:
$v \geq 9$

10) Step3: Combine OR solutions

Solution: $v \leq 0$ or $v \geq 9$
---

11) Step1: Solve $2+5n < 9n-10$

Subtract 2, subtract $5n$:
$12 < 4n \implies n > 3$

11) Step2: Solve $-3n-8 > 10n-8$

Add 8, add $3n$:
$0 > 13n \implies n < 0$

11) Step3: Combine OR solutions

Solution: $n < 0$ or $n > 3$
---

12) Step1: Solve $7r-1 \leq 8r-6$

Add 1, subtract $7r$:
$0 \leq r - 5 \implies r \geq 5$

12) Step2: Solve $r+4 \geq 6r+9$

Subtract 4, subtract $6r$:
$-5r \geq 5 \implies r \leq -1$

12) Step3: Combine OR solutions

Solution: $r \leq -1$ or $r \geq 5$
---

13) Step1: Solve $6+7n > 5n-6$

Subtract 6, subtract $5n$:
$2n > -12 \implies n > -6$

13) Step2: Solve $6n-5 > 9+8n$

Add 5, subtract $6n$:
$0 > 14 + 2n \implies n < -7$

13) Step3: Combine OR solutions

Solution: $n < -7$ or $n > -6$
---

14) Step1: Solve $7n+6 > 6-n$

Subtract 6, add $n$:
$8n > 0 \implies n > 0$

14) Step2: Solve $-7+9n > 10n-1$

Add 7, subtract $10n$:
$-n > 6 \implies n < -6$

14) Step3: Combine OR solutions

Solution: $n < -6$ or $n > 0$
---

15) Step1: Solve $-2-n \geq -5n+10$

Add 2, add $5n$:
$4n \geq 12 \implies n \geq 3$

15) Step2: Solve $2-3n < -4n+4$

Subtract 2, add $4n$:
$n < 2$

15) Step3: Combine OR solutions

Solution: $n < 2$ or $n \geq 3$
---

16) Step1: Solve $5x+2 > 9x+2$

Subtract 2, subtract $5x$:
$0 > 4x \implies x < 0$

16) Step2: Solve $8x+7 \leq 2+9x$

Subtract 7, subtract $8x$:
$0 \leq x - 5 \implies x \geq 5$

16) Step3: Combine OR solutions

Solution: $x < 0$ or $x \geq 5$
---

17) Step1: Solve $3n+10 < 5+2n$

Subtract 10, subtract $2n$:
$n < -5$

17) Step2: Solve $5+8n \geq -1+6n$

Subtract 5, subtract $6n$:
$2n \geq -6 \implies n \geq -3$

17) Step3: Combine OR solutions

Solution: $n < -5$ or $n \geq -3$
---

18) Step1: Solve $1-4x < -3x-3$

Subtract 1, add $4x$:
$0 < x - 4 \implies x > 4$

18) Step2: Solve $x+8 > 8x-6$

Subtract 8, subtract $8x$:
$-7x > -14 \implies x < 2$

18) Step3: Combine OR solutions

Solution: $x < 2$ or $x > 4$
---

19) Step1: Solve $5m+7 < 6m-8$

Subtract 7, subtract $5m$:
$0 < m - 15 \implies m > 15$

19) Step2: Solve $-2+2m \geq 3m-10$

Add 2, subtract $3m$:
$-m \geq -8 \implies m \leq 8$

19) Step3: Combine OR solutions

Solution: $m \leq 8$ or $m > 15$
---

20) Step1: Solve $5+9n \geq 5-7n$

Subtract 5, add $7n$:
$16n \geq 0 \implies n \geq 0$

20) Step2: Solve $7n+1 \leq 6n-6$

Subtract 1, subtract $6n$:
$n \leq -7$

20) Step3: Combine OR solutions

Solution: $n \leq -7$ or $n \geq 0$

Answer:

1. $x \leq -15$ or $x \geq 0$
2. $r \leq -8$ or $r > 1$
3. $b < -3$ or $b \geq 0$
4. $m < -15$ or $m > -3$
5. $v < 0$ or $v \geq 8$
6. $p < 1$ or $p > 9$
7. $x < 0$ or $x \geq 7$
8. $b < -5$ or $b \geq -1$
9. $b \leq -14$ or $b \geq -4$
10. $v \leq 0$ or $v \geq 9$
11. $n < 0$ or $n > 3$
12. $r \leq -1$ or $r \geq 5$
13. $n < -7$ or $n > -6$
14. $n < -6$ or $n > 0$
15. $n < 2$ or $n \geq 3$
16. $x < 0$ or $x \geq 5$
17. $n < -5$ or $n \geq -3$
18. $x < 2$ or $x > 4$
19. $m \leq 8$ or $m > 15$
20. $n \leq -7$ or $n \geq 0$

(Graphing instructions: For each solution, plot an open circle at exclusive endpoints, closed circle at inclusive endpoints, and shade the regions matching the inequality direction.)