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Solve for $x$ . $\newline$ $(x - 2)(x - 3) \leq 0$ $\newline$ Write a compound inequality like $1 < x < 3$ or like $x < 1$ or $x > 3$ .______

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Solve quadratic inequalities

Full solution

Q. Solve for

x

.

\newline

(x - 2)(x - 3) \leq 0

\newline

Write a compound inequality like

1 < x < 3

or like

x < 1

or

x > 3

.______

Find Critical Points: Find the critical points of $(x - 2)(x - 3)$ . $(x - 2)(x - 3) = 0$ $x - 2 = 0$ implies $x = 2$ . $x - 3 = 0$ implies $x = 3$ .Critical points: $2, 3$ .
Identify Intervals: Identify the intervals using the critical points. $\newline$ $2$ and $3$ divide the number line into three parts. $\newline$ Intervals: $(-\infty, 2$ ), $(2, 3)$ , $(3, \infty)$ .
Test Sign $(-\infty, 2)$ : Test the sign of $(x - 2)(x - 3)$ over $(-\infty, 2)$ . $\newline$ Pick $x = 1$ (which is in the interval). $\newline$ Sign of $(x - 2)$ : -. $\newline$ Sign of $(x - 3)$ : -. $\newline$ Sign of $(x - 2)(x - 3)$ : $(-)(-) = +$ .
Test Sign $(2, 3)$ : Test the sign of $(x - 2)(x - 3)$ over $(2, 3)$ . $\newline$ Pick $x = 2.5$ (which is in the interval). $\newline$ Sign of $(x - 2)$ : +. $\newline$ Sign of $(x - 3)$ : -. $\newline$ Sign of $(x - 2)(x - 3)$ : $(+)(-) = -$ .
Test Sign \(3, \infty): Test the sign of x - \(2)(x - $3$ ) over \(3, \infty). $\newline$ Pick x = \(4) (which is in the interval). $\newline$ Sign of x - \(2): +. $\newline$ Sign of x - \(3): +. $\newline$ Sign of x - \(2)(x - $3$ ): (+)(+) = +.
Compound Inequality: $(x - 2)(x - 3) \leq 0$ . Find the solution as a compound inequality. $(x - 2)(x - 3)$ is non-positive over $[2, 3]$ . Compound inequality: $2 \leq x \leq 3$ .

More problems from Solve quadratic 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

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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