Question

solve each inequality. graph the solutions.

1. 8 < x + 2
2. 7x < - 49
3. $\frac{x}{5}leq - 6$
4. x - 6 > - 3
5. 11 + x < 4
6. 4xleq20
7. is each number a solution of y + 8leq5?

-3__ 0 $\frac{-13}{4}$ -2.9__

1. is each number a solution of -7x < -21?

3__ 0 $\frac{8}{3}$ 4__
write an inequality.

1. <graph with -3, -2, -1, 0, 1, 2, 3 marked and an open - circle at 0 and arrow to the left>
2. <graph with -3, -2, -1, 0, 1, 2, 3 marked and a closed - circle at 0 and arrow to the left>
3. the charity hopes to raise more than $10,000.
4. no more than 120 people are allowed in the gym at one time.
5. to train for a marathon, a runner decides that she must run at least 12 miles a day.

write and solve an inequality to model each situation.

1. eight less than a number x is at least 17.
2. you earn $7.25 per hour baby - sitting. write and solve an inequality to find how many full hours you must work to earn at least $200.

Explanation:

Step1: Solve 8 < x + 2

Subtract 2 from both sides.
$8-2$6 6$

Step2: Solve 7x < - 49

Divide both sides by 7.
$\frac{7x}{7}<\frac{-49}{7}$
$x<-7$

Step3: Solve $\frac{x}{5}\leq - 6$

Multiply both sides by 5.
$\frac{x}{5}\times5\leq - 6\times5$
$x\leq - 30$

Step4: Solve x - 6 > - 3

Add 6 to both sides.
$x-6 + 6>-3 + 6$
$x>3$

Step5: Solve 11 + x < 4

Subtract 11 from both sides.
$11 + x-11<4-11$
$x<-7$

Step6: Solve 4x ≤ 20

Divide both sides by 4.
$\frac{4x}{4}\leq\frac{20}{4}$
$x\leq5$

Step7: Check solutions for y + 8 ≤ 5

For $y=-3$: $-3 + 8=5$, so $y=-3$ is a solution.
For $y = 0$: $0+8=8>5$, so $y = 0$ is not a solution.
For $y=-\frac{13}{4}=-3.25$: $-3.25 + 8 = 4.75\leq5$, so $y=-\frac{13}{4}$ is a solution.
For $y=-2.9$: $-2.9 + 8=5.1>5$, so $y=-2.9$ is not a solution.

Step8: Check solutions for - 7x < - 21

Divide both sides by - 7 and reverse the inequality sign. $x>3$
For $x = 3$: $-7\times3=-21$, not less than - 21, so $x = 3$ is not a solution.
For $x = 0$: $-7\times0=0>-21$, so $x = 0$ is not a solution.
For $x=\frac{8}{3}\approx2.67<3$, so $x=\frac{8}{3}$ is not a solution.
For $x = 4$: $-7\times4=-28<-21$, so $x = 4$ is a solution.

Step9: Write inequality from graph

The graph has an open - circle at - 1 and the line goes to the right, so the inequality is $x>-1$

Step10: Write inequality from graph

The graph has a closed - circle at 0 and the line goes to the left, so the inequality is $x\leq0$

Step11: Write inequality for charity

Let $a$ be the amount raised. The inequality is $a>10000$

Step12: Write inequality for gym

Let $p$ be the number of people. The inequality is $p\leq120$

Step13: Write inequality for runner

Let $d$ be the number of miles run per day. The inequality is $d\geq12$

Step14: Write and solve inequality

The inequality is $x - 8\geq17$
Add 8 to both sides: $x-8 + 8\geq17+8$
$x\geq25$

Step15: Write and solve inequality

Let $h$ be the number of hours worked. The inequality is $7.25h\geq200$
Divide both sides by 7.25: $h\geq\frac{200}{7.25}=\frac{200\times4}{7.25\times4}=\frac{800}{29}\approx27.59$
Since we need full - hours, $h\geq28$

Answer:

1. $x>6$
2. $x<-7$
3. $x\leq - 30$
4. $x>3$
5. $x<-7$
6. $x\leq5$
7. - 3: Yes, 0: No, $-\frac{13}{4}$: Yes, - 2.9: No
8. 3: No, 0: No, $\frac{8}{3}$: No, 4: Yes
9. $x>-1$
10. $x\leq0$
11. $a>10000$
12. $p\leq120$
13. $d\geq12$
14. $x\geq25$
15. $h\geq28$