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Math  /  Algebra

QuestionIn parts (a)-(f), use the given figure. (a) Solve f(x)=24f(x)=24. (b) Solve f(x)=45f(x)=45. (c) Solve f(x)=0f(x)=0. (d) Solve f(x)>24f(x)>24. (e) Solve f(x)≤45f(x) \leq 45. (f) Solve 0<f(x)<450<f(x)<45. (a) For what value of x does f(x)=24\mathrm{f}(\mathrm{x})=24 ? x=□\mathrm{x}=\square (b) For what value of xx does f(x)=45f(x)=45 ? x=□\mathrm{x}=\square □\square (c) For what value of xx does f(x)=0f(x)=0 ? x=□\mathrm{x}=\square (d) For which values of x is f(x)>24\mathrm{f}(\mathrm{x})>24 ?
For every x in the interval □,f(x)>24\square, \mathrm{f}(\mathrm{x})>24. □\square (Type your answer in interval notation.) (e) For which values of x is f(x)≤45\mathrm{f}(\mathrm{x}) \leq 45 ?
For every x in the interval □,f(x)≤45\square, \mathrm{f}(\mathrm{x}) \leq 45. □\square (Type your answer in interval notation.) (f) For which values of x is 0<f(x)<450<\mathrm{f}(\mathrm{x})<45 ?

Studdy Solution

STEP 1

1. The function f(x) f(x) is a linear function as it is represented by a straight line.
2. The points given are (−20,0)(-20, 0), (20,24)(20, 24), and (55,45)(55, 45).
3. We need to solve for specific values and conditions of f(x) f(x) using the graph.

STEP 2

1. Determine the equation of the line f(x) f(x) .
2. Solve f(x)=24 f(x) = 24 .
3. Solve f(x)=45 f(x) = 45 .
4. Solve f(x)=0 f(x) = 0 .
5. Solve f(x)>24 f(x) > 24 .
6. Solve f(x)≤45 f(x) \leq 45 .
7. Solve 0<f(x)<45 0 < f(x) < 45 .

STEP 3

Determine the equation of the line using two points. Use the points (20,24)(20, 24) and (55,45)(55, 45).
Calculate the slope m m :
m=45−2455−20=2135=35 m = \frac{45 - 24}{55 - 20} = \frac{21}{35} = \frac{3}{5}
Use the point-slope form y−y1=m(x−x1) y - y_1 = m(x - x_1) with point (20,24)(20, 24):
y−24=35(x−20) y - 24 = \frac{3}{5}(x - 20)
Simplify to find the equation of the line:
y=35x+12 y = \frac{3}{5}x + 12
Thus, f(x)=35x+12 f(x) = \frac{3}{5}x + 12 .

STEP 4

Solve f(x)=24 f(x) = 24 :
35x+12=24 \frac{3}{5}x + 12 = 24
Subtract 12 from both sides:
35x=12 \frac{3}{5}x = 12
Multiply both sides by 53\frac{5}{3}:
x=20 x = 20

STEP 5

Solve f(x)=45 f(x) = 45 :
35x+12=45 \frac{3}{5}x + 12 = 45
Subtract 12 from both sides:
35x=33 \frac{3}{5}x = 33
Multiply both sides by 53\frac{5}{3}:
x=55 x = 55

STEP 6

Solve f(x)=0 f(x) = 0 :
35x+12=0 \frac{3}{5}x + 12 = 0
Subtract 12 from both sides:
35x=−12 \frac{3}{5}x = -12
Multiply both sides by 53\frac{5}{3}:
x=−20 x = -20

STEP 7

Solve f(x)>24 f(x) > 24 :
From the equation 35x+12>24 \frac{3}{5}x + 12 > 24 , solve for x x :
35x>12 \frac{3}{5}x > 12
Multiply both sides by 53\frac{5}{3}:
x>20 x > 20
Thus, the interval is (20,∞) (20, \infty) .

STEP 8

Solve f(x)≤45 f(x) \leq 45 :
From the equation 35x+12≤45 \frac{3}{5}x + 12 \leq 45 , solve for x x :
35x≤33 \frac{3}{5}x \leq 33
Multiply both sides by 53\frac{5}{3}:
x≤55 x \leq 55
Thus, the interval is (−∞,55] (-\infty, 55] .

STEP 9

Solve 0<f(x)<45 0 < f(x) < 45 :
From the equation 0<35x+12<45 0 < \frac{3}{5}x + 12 < 45 , solve for x x :
First, solve 0<35x+12 0 < \frac{3}{5}x + 12 :
35x>−12 \frac{3}{5}x > -12
Multiply both sides by 53\frac{5}{3}:
x>−20 x > -20
Next, solve 35x+12<45 \frac{3}{5}x + 12 < 45 :
35x<33 \frac{3}{5}x < 33
Multiply both sides by 53\frac{5}{3}:
x<55 x < 55
Thus, the interval is (−20,55) (-20, 55) .
The solutions are: (a) x=20 x = 20 (b) x=55 x = 55 (c) x=−20 x = -20 (d) (20,∞) (20, \infty) (e) (−∞,55] (-\infty, 55] (f) (−20,55) (-20, 55)

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