Math / Algebra

QuestionSuppose that $f(x)=6 x-1$ and $g(x)=-3 x+8$ (a) Solve $f(x)=0$ . (c) Solve $f(x)=g(x)$ . (b) Solve $f(x)>0$ . (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 $f(x)=0$ ? $x=\frac{1}{6}$ (Type an integer or a simplified fraction.) (b) For which values of $x$ is $f(x)>0$ ? $\square$ (Type your answer in interval notation. Use integers or fractions for any numbers in the expression.)

Studdy Solution

STEP 1

1. We are given two functions $f(x) = 6x - 1$ and $g(x) = -3x + 8$ .
2. We need to solve different conditions involving these functions: $f(x) = 0$ , $f(x) = g(x)$ , and $f(x) > 0$ .
3. We will also graph the functions to find the intersection point visually.

STEP 2

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

STEP 3

To solve $f(x) = 0$ , set the function equal to zero:
$6x - 1 = 0$

STEP 4

Add 1 to both sides of the equation:
$6x = 1$

STEP 5

Divide both sides by 6 to solve for $x$ :
$x = \frac{1}{6}$

STEP 6

To solve $f(x) = g(x)$ , set the two functions equal to each other:
$6x - 1 = -3x + 8$

STEP 7

Add $3x$ to both sides:
$6x + 3x - 1 = 8$

STEP 8

Combine like terms:
$9x - 1 = 8$

STEP 9

Add 1 to both sides:
$9x = 9$

STEP 10

Divide both sides by 9:
$x = 1$

STEP 11

To solve $f(x) > 0$ , set the inequality:
$6x - 1 > 0$

STEP 12

Add 1 to both sides:
$6x > 1$

STEP 13

Divide both sides by 6:
$x > \frac{1}{6}$

STEP 14

Graph $y = f(x)$ and $y = g(x)$ . The point of intersection is the solution to $f(x) = g(x)$ .
The intersection point is $(1, 5)$ .
Solution Summary: (a) $x = \frac{1}{6}$ (b) $x > \frac{1}{6}$ (Interval notation: $\left(\frac{1}{6}, \infty\right)$ ) (c) $x = 1$ (e) Intersection point: $(1, 5)$

Was this helpful?

Studdy Solution

STEP 1

1. We are given two functions $f(x) = 6x - 1$ and $g(x) = -3x + 8$ .
2. We need to solve different conditions involving these functions: $f(x) = 0$ , $f(x) = g(x)$ , and $f(x) > 0$ .
3. We will also graph the functions to find the intersection point visually.

STEP 2

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

STEP 3

To solve $f(x) = 0$ , set the function equal to zero:
$6x - 1 = 0$

STEP 4

Add 1 to both sides of the equation:
$6x = 1$

STEP 5

Divide both sides by 6 to solve for $x$ :
$x = \frac{1}{6}$

STEP 6

To solve $f(x) = g(x)$ , set the two functions equal to each other:
$6x - 1 = -3x + 8$

STEP 7

Add $3x$ to both sides:
$6x + 3x - 1 = 8$

STEP 8

Combine like terms:
$9x - 1 = 8$

STEP 9

Add 1 to both sides:
$9x = 9$

STEP 10

Divide both sides by 9:
$x = 1$

STEP 11

To solve $f(x) > 0$ , set the inequality:
$6x - 1 > 0$

STEP 12

Add 1 to both sides:
$6x > 1$

STEP 13

Divide both sides by 6:
$x > \frac{1}{6}$

STEP 14

Graph $y = f(x)$ and $y = g(x)$ . The point of intersection is the solution to $f(x) = g(x)$ .
The intersection point is $(1, 5)$ .
Solution Summary: (a) $x = \frac{1}{6}$ (b) $x > \frac{1}{6}$ (Interval notation: $\left(\frac{1}{6}, \infty\right)$ ) (c) $x = 1$ (e) Intersection point: $(1, 5)$

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Studdy Solution

STEP 1

STEP 2

1. Solve f(x)=0 f(x) = 0 f(x)=0.2. Solve f(x)=g(x) f(x) = g(x) f(x)=g(x).3. Solve f(x)>0 f(x) > 0 f(x)>0.4. 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

To solve f(x)=0 f(x) = 0 f(x)=0, set the function equal to zero:6x−1=0 6x - 1 = 0 6x−1=0

STEP 4

Add 1 to both sides of the equation:6x=1 6x = 1 6x=1

STEP 5

Divide both sides by 6 to solve for x x x:x=16 x = \frac{1}{6} x=61​

STEP 6

To solve f(x)=g(x) f(x) = g(x) f(x)=g(x), set the two functions equal to each other:6x−1=−3x+8 6x - 1 = -3x + 8 6x−1=−3x+8

STEP 7

Add 3x 3x 3x to both sides:6x+3x−1=8 6x + 3x - 1 = 8 6x+3x−1=8

STEP 8

Combine like terms:9x−1=8 9x - 1 = 8 9x−1=8

STEP 9

Add 1 to both sides:9x=9 9x = 9 9x=9

STEP 10

Divide both sides by 9:x=1 x = 1 x=1

STEP 11

To solve f(x)>0 f(x) > 0 f(x)>0, set the inequality:6x−1>0 6x - 1 > 0 6x−1>0

STEP 12

Add 1 to both sides:6x>1 6x > 1 6x>1

STEP 13

Divide both sides by 6:x>16 x > \frac{1}{6} x>61​

STEP 14

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

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

To solve $f(x) = 0$ , set the function equal to zero:
$6x - 1 = 0$

Add 1 to both sides of the equation:
$6x = 1$

Divide both sides by 6 to solve for $x$ :
$x = \frac{1}{6}$

To solve $f(x) = g(x)$ , set the two functions equal to each other:
$6x - 1 = -3x + 8$

Add $3x$ to both sides:
$6x + 3x - 1 = 8$

Combine like terms:
$9x - 1 = 8$

Add 1 to both sides:
$9x = 9$

Divide both sides by 9:
$x = 1$

To solve $f(x) > 0$ , set the inequality:
$6x - 1 > 0$

Add 1 to both sides:
$6x > 1$

Divide both sides by 6:
$x > \frac{1}{6}$