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Solve for $z$ . $\newline$ $|-8z| \geq 8$ $\newline$ $\newline$ Write a compound inequality like $1 < x < 3$ or like $x < 1$ or $x > 3$ . Use integers, proper fractions, or improper fractions in simplest form. $\newline$ ______

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Solve absolute value inequalities

Full solution

Q. Solve for

z

.

\newline

|-8z| \geq 8

\newline

\newline

Write a compound inequality like

1 < x < 3

or like

x < 1

or

x > 3

. Use integers, proper fractions, or improper fractions in simplest form.

\newline

______

Given Inequality: We have the inequality: $\newline$ $|-8z| \geq 8$ $\newline$ First, we solve for $|8z|$ . $\newline$ $|-8z| \geq 8$ $\newline$ This means that the absolute value of $-8z$ is greater than or equal to $8$ .
Splitting Inequality: The absolute value inequality $|-8z| \geq 8$ can be split into two separate inequalities because if the expression inside the absolute value is non-negative, it is equal to the expression itself, and if it is negative, it is equal to the negative of the expression. Therefore, we have: $\newline$ $-8z \geq 8$ or $-8z \leq -8$
Solving First Inequality: Now we solve each inequality separately. Starting with the first one: $\newline$ $-8z \geq 8$ $\newline$ Divide both sides by $-8$ , remembering to reverse the inequality sign because we are dividing by a negative number: $\newline$ $z \leq -1$
Solving Second Inequality: Now we solve the second inequality: $\newline$ $-8z \leq -8$ $\newline$ Divide both sides by $-8$ : $\newline$ $z \geq 1$
Combining Solutions: Combining both parts of the solution, we get the compound inequality: $\newline$ $z \leq -1$ or $z \geq 1$ $\newline$ This is the final answer in the form of a compound inequality.

More problems from Solve absolute value inequalities

Question

s

\newline

|s| + 3 \geq 8

\newline

\newline

Write a compound inequality like

1 < x < 3

x < 1

x > 3

. Use integers, proper fractions, or improper fractions in simplest form.

\newline

\newline

Posted 1 month ago

Question

x

\newline

(x - 1)(x + 1) > 0

\newline

\newline

Write a compound inequality like

1 < x < 3

x < 1

x > 3

\newline

Posted 1 month ago

Question

A particular company charges advertisers a one time cost of

\$500

, in addition to

\$4.50

for every one thousand times an advertisement is shown on the company's webpage. An advertiser wants its ad to appear

M

thousand times on the webpage, but does not want to spend more than

\$5,000

. Which of the following inequalities best describes the situation?

\newline

\newline

500+4.50M\geq5,000

\newline

4.50+500M>5,000

\newline

500+4.50M\leq5,000

\newline

500+4.50M<5,000

Posted 1 month ago

Question

7x+1 < x+9

\newline

Which of the following best describes the solutions to the inequality shown?

\newline

1

\newline

x > \frac{4}{3}

\newline

x < \frac{4}{3}

\newline

x < 1

\newline

x < \frac{5}{3}

Posted 4 months ago

Question

52-3x < -14

\newline

Which of the following best describes the solutions to the inequality shown?

\newline

1

\newline

x > -\frac{38}{3}

\newline

x > \frac{38}{3}

\newline

x < 22

\newline

x > 22

Posted 4 months ago

Question

5-3x > 2x+2

\newline

Which of the following best describes the solutions to the inequality shown?

\newline

1

\newline

x < \frac{7}{5}

\newline

x < 3

\newline

x < \frac{3}{5}

\newline

x > \frac{3}{5}

Posted 4 months ago

Question

Rani is a real estate agent. For each house she sells, she pays

\$ 100

in fees, but earns a commission of

1.8 \%

of the selling price of the house. Rani's total profit from a particular house is

\$ 4,580

p

represents the selling price of the house, which equation best models the situation?

\newline

1

\newline

0.018 p-100=4580

\newline

0.018 p+100=4580

\newline

(100-0.018) p=4580

\newline

(100+0.018) p=4580

Posted 1 month ago

Question

A commercial airplane that is

1

500

2

500

-mile journey is traveling at

450

knots in still air when it picks up a tailwind of

150

knots (in the same direction). If

h

is the number of hours remaining in the airplane's flight, which of the following equations best describes the situation?

\newline

1

\mathrm{knot}=1.15

miles per hour (mph)

\newline

1

\newline

1,500+690 h=2,500

\newline

1,500+600 h=2,500

\newline

1,500-600 h=2,500

\newline

1,500-690 h=2,500

Posted 26 days ago

Question

A weeping willow that is

15

feet in height grows to a maximum height of

35

y

years at a constant rate of

24

inches per year. Which of the following equations best describes this situation?

\newline

1

=12

\newline

1

\newline

35=15+2 y

\newline

35=15+24 y

\newline

35=15-2 y

\newline

35=15-24 y

Posted 4 months ago

Question

The gas mileage for a car is

23

miles per gallon when the car travels at

60

miles per hour. The car begins a trip with

13

gallons in its tank, travels at an average speed of

60

miles per hour for

h

hours, and ends the trip with

10

gallons in its tank. Which of the following equations best models this situation?

\newline

1

\newline

13-\frac{23 h}{60}=10

\newline

13-\frac{60 h}{23}=10

\newline

\frac{13-60 h}{23}=10

\newline

\frac{13-23 h}{60}=10

Posted 2 months ago