Question

infinite algebra 1
multi - step inequalities
solve each inequality and graph its solution.

1. (3geq - 5n+2n)
2. (6x + 2+6xlt14)
3. (-p - 4pgt - 10)
4. (18geq3k + 4k)
5. (9geq - 2m+2 - 3)
6. (-3-6(4x + 6)gt - 111)
7. (6-4(6n + 7)geq122)
8. (-138geq - 6(6b - 7))
9. (167lt6 + 7(2 - 7n))
10. (5(6 + 3r)+7geq127)
11. (-8x + 2x-16lt - 5x+7x)
12. (-1-6x - 6gt - 11-7x)

Explanation:

1) Step1: Rearrange like terms

$3 < -5n + 2n$

1) Step2: Combine like terms

$3 < -3n$

1) Step3: Divide by -3, flip inequality

$\frac{3}{-3} > n$
$n < -1$
---

2) Step1: Combine like terms

$12x + 2 < 14$

2) Step2: Subtract 2 from both sides

$12x < 14 - 2$
$12x < 12$

2) Step3: Divide by 12

$x < \frac{12}{12}$
$x < 1$
---

3) Step1: Combine like terms

$-5p > -10$

3) Step2: Divide by -5, flip inequality

$p < \frac{-10}{-5}$
$p < 2$
---

4) Step1: Rearrange like terms

$18 \geq 7k$

4) Step2: Divide by 7

$k \leq \frac{18}{7} \approx 2.57$
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5) Step1: Subtract 2 from both sides

$9 - 2 \geq -2m - 3$
$7 \geq -2m - 3$

5) Step2: Add 3 to both sides

$7 + 3 \geq -2m$
$10 \geq -2m$

5) Step3: Divide by -2, flip inequality

$\frac{10}{-2} \leq m$
$m \geq -5$
---

6) Step1: Expand the bracket

$-3 -24x -36 > -111$

6) Step2: Combine constants

$-24x -39 > -111$

6) Step3: Add 39 to both sides

$-24x > -111 + 39$
$-24x > -72$

6) Step4: Divide by -24, flip inequality

$x < \frac{-72}{-24}$
$x < 3$
---

7) Step1: Expand the bracket

$6 -24m -28 \geq 122$

7) Step2: Combine constants

$-24m -22 \geq 122$

7) Step3: Add 22 to both sides

$-24m \geq 122 + 22$
$-24m \geq 144$

7) Step4: Divide by -24, flip inequality

$m \leq \frac{144}{-24}$
$m \leq -6$
---

8) Step1: Expand the bracket

$-138 \geq -36b + 42$

8) Step2: Subtract 42 from both sides

$-138 -42 \geq -36b$
$-180 \geq -36b$

8) Step3: Divide by -36, flip inequality

$\frac{-180}{-36} \leq b$
$b \geq 5$
---

9) Step1: Expand the bracket

$167 < 6 + 14 - 49r$

9) Step2: Combine constants

$167 < 20 - 49r$

9) Step3: Subtract 20 from both sides

$167 -20 < -49r$
$147 < -49r$

9) Step4: Divide by -49, flip inequality

$\frac{147}{-49} > r$
$r < -3$
---

10) Step1: Expand the bracket

$30 + 15r + 7 \geq 127$

10) Step2: Combine constants

$15r + 37 \geq 127$

10) Step3: Subtract 37 from both sides

$15r \geq 127 - 37$
$15r \geq 90$

10) Step4: Divide by 15

$r \geq \frac{90}{15}$
$r \geq 6$
---

11) Step1: Combine like terms

$-6x -16 < 2x$

11) Step2: Rearrange like terms

$-16 < 2x + 6x$
$-16 < 8x$

11) Step3: Divide by 8

$\frac{-16}{8} < x$
$x > -2$
---

12) Step1: Combine like terms

$-6x -7 > -11 -7x$

12) Step2: Rearrange like terms

$-6x +7x > -11 +7$

12) Step3: Simplify both sides

$x > -4$

Answer:

1. $n < -1$ (Graph: Open circle at -1, arrow left)
2. $x < 1$ (Graph: Open circle at 1, arrow left)
3. $p < 2$ (Graph: Open circle at 2, arrow left)
4. $k \leq \frac{18}{7}$ (Graph: Closed circle at $\frac{18}{7}$, arrow left)
5. $m \geq -5$ (Graph: Closed circle at -5, arrow right)
6. $x < 3$ (Graph: Open circle at 3, arrow left)
7. $m \leq -6$ (Graph: Closed circle at -6, arrow left)
8. $b \geq 5$ (Graph: Closed circle at 5, arrow right)
9. $r < -3$ (Graph: Open circle at -3, arrow left)
10. $r \geq 6$ (Graph: Closed circle at 6, arrow right)
11. $x > -2$ (Graph: Open circle at -2, arrow right)
12. $x > -4$ (Graph: Open circle at -4, arrow right)