Math / Algebra

QuestionSuppose that $f(x)=6 x-1$ and $g(x)=-3 x+8$ (a) Solve $f(x)=0$ . (b) Solve $f(x)>0$ . (c) Solve $f(x)=g(x)$ . (d) Solve $f(x) \leq g(x)$ . (e) Graph $y=f(x)$ and $y=g(x)$ and find the point that represents the solution to the equation $f(x)=g(x)$ . (a) For what value of $x$ does $f(x)=0$ ? $x=\square$ (Type an integer or a simplified fraction.)

Studdy Solution

STEP 1

1. We are given two functions: $f(x) = 6x - 1$ and $g(x) = -3x + 8$ .
2. We need to solve various equations and inequalities involving these functions.
3. We will graph the functions to find the intersection point.
4. We will express solutions in terms of integers or simplified fractions.

STEP 2

1. Solve $f(x) = 0$ .
2. Solve $f(x) > 0$ .
3. Solve $f(x) = g(x)$ .
4. Solve $f(x) \leq g(x)$ .
5. Graph $y = f(x)$ and $y = g(x)$ and find the intersection point.

STEP 3

Solve $f(x) = 0$ .
Set the function equal to zero:
$6x - 1 = 0$
Add 1 to both sides:
$6x = 1$
Divide both sides by 6:
$x = \frac{1}{6}$

STEP 4

Solve $f(x) > 0$ .
Set up the inequality:
$6x - 1 > 0$
Add 1 to both sides:
$6x > 1$
Divide both sides by 6:
$x > \frac{1}{6}$

STEP 5

Solve $f(x) = g(x)$ .
Set the functions equal to each other:
$6x - 1 = -3x + 8$
Add $3x$ to both sides:
$9x - 1 = 8$
Add 1 to both sides:
$9x = 9$
Divide both sides by 9:
$x = 1$

STEP 6

Solve $f(x) \leq g(x)$ .
Set up the inequality:
$6x - 1 \leq -3x + 8$
Add $3x$ to both sides:
$9x - 1 \leq 8$
Add 1 to both sides:
$9x \leq 9$
Divide both sides by 9:
$x \leq 1$

STEP 7

Graph $y = f(x)$ and $y = g(x)$ and find the intersection point.
- Plot the line $y = 6x - 1$ . - Plot the line $y = -3x + 8$ . - The intersection point of these lines is the solution to $f(x) = g(x)$ .
The intersection occurs at $x = 1$ .
Solution to (a): For what value of $x$ does $f(x) = 0$ ?
$x = \frac{1}{6}$

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Studdy Solution

STEP 1

1. We are given two functions: $f(x) = 6x - 1$ and $g(x) = -3x + 8$ .
2. We need to solve various equations and inequalities involving these functions.
3. We will graph the functions to find the intersection point.
4. We will express solutions in terms of integers or simplified fractions.

STEP 2

1. Solve $f(x) = 0$ .
2. Solve $f(x) > 0$ .
3. Solve $f(x) = g(x)$ .
4. Solve $f(x) \leq g(x)$ .
5. Graph $y = f(x)$ and $y = g(x)$ and find the intersection point.

STEP 3

Solve $f(x) = 0$ .
Set the function equal to zero:
$6x - 1 = 0$
Add 1 to both sides:
$6x = 1$
Divide both sides by 6:
$x = \frac{1}{6}$

STEP 4

Solve $f(x) > 0$ .
Set up the inequality:
$6x - 1 > 0$
Add 1 to both sides:
$6x > 1$
Divide both sides by 6:
$x > \frac{1}{6}$

STEP 5

Solve $f(x) = g(x)$ .
Set the functions equal to each other:
$6x - 1 = -3x + 8$
Add $3x$ to both sides:
$9x - 1 = 8$
Add 1 to both sides:
$9x = 9$
Divide both sides by 9:
$x = 1$

STEP 6

Solve $f(x) \leq g(x)$ .
Set up the inequality:
$6x - 1 \leq -3x + 8$
Add $3x$ to both sides:
$9x - 1 \leq 8$
Add 1 to both sides:
$9x \leq 9$
Divide both sides by 9:
$x \leq 1$

STEP 7

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Studdy Solution

STEP 1

STEP 2

1. Solve f(x)=0 f(x) = 0 f(x)=0.2. Solve f(x)>0 f(x) > 0 f(x)>0.3. Solve f(x)=g(x) f(x) = g(x) f(x)=g(x).4. Solve f(x)≤g(x) f(x) \leq g(x) f(x)≤g(x).5. Graph y=f(x) y = f(x) y=f(x) and y=g(x) y = g(x) y=g(x) and find the intersection point.

STEP 3

Solve f(x)=0 f(x) = 0 f(x)=0.Set the function equal to zero:6x−1=0 6x - 1 = 0 6x−1=0Add 1 to both sides:6x=1 6x = 1 6x=1Divide both sides by 6:x=16 x = \frac{1}{6} x=61​

STEP 4

Solve f(x)>0 f(x) > 0 f(x)>0.Set up the inequality:6x−1>0 6x - 1 > 0 6x−1>0Add 1 to both sides:6x>1 6x > 1 6x>1Divide both sides by 6:x>16 x > \frac{1}{6} x>61​

STEP 5

Solve f(x)=g(x) f(x) = g(x) f(x)=g(x).Set the functions equal to each other:6x−1=−3x+8 6x - 1 = -3x + 8 6x−1=−3x+8Add 3x 3x 3x to both sides:9x−1=8 9x - 1 = 8 9x−1=8Add 1 to both sides:9x=9 9x = 9 9x=9Divide both sides by 9:x=1 x = 1 x=1

STEP 6

Solve f(x)≤g(x) f(x) \leq g(x) f(x)≤g(x).Set up the inequality:6x−1≤−3x+8 6x - 1 \leq -3x + 8 6x−1≤−3x+8Add 3x 3x 3x to both sides:9x−1≤8 9x - 1 \leq 8 9x−1≤8Add 1 to both sides:9x≤9 9x \leq 9 9x≤9Divide both sides by 9:x≤1 x \leq 1 x≤1

STEP 7

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1. Solve $f(x) = 0$ .
2. Solve $f(x) > 0$ .
3. Solve $f(x) = g(x)$ .
4. Solve $f(x) \leq g(x)$ .
5. Graph $y = f(x)$ and $y = g(x)$ and find the intersection point.

Solve $f(x) = 0$ .
Set the function equal to zero:
$6x - 1 = 0$
Add 1 to both sides:
$6x = 1$
Divide both sides by 6:
$x = \frac{1}{6}$

Solve $f(x) > 0$ .
Set up the inequality:
$6x - 1 > 0$
Add 1 to both sides:
$6x > 1$
Divide both sides by 6:
$x > \frac{1}{6}$

Solve $f(x) = g(x)$ .
Set the functions equal to each other:
$6x - 1 = -3x + 8$
Add $3x$ to both sides:
$9x - 1 = 8$
Add 1 to both sides:
$9x = 9$
Divide both sides by 9:
$x = 1$

Solve $f(x) \leq g(x)$ .
Set up the inequality:
$6x - 1 \leq -3x + 8$
Add $3x$ to both sides:
$9x - 1 \leq 8$
Add 1 to both sides:
$9x \leq 9$
Divide both sides by 9:
$x \leq 1$