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 $f(x)=0$ ? $x=\frac{1}{6}$ (Type an integer or a simplified fraction.) (b) For which values of x is $\mathrm{f}(\mathrm{x})>0$ ? $\left(\frac{1}{6}, \infty\right)$ (Type your answer in interval notation. Use integers or fractions for any numbers in the expression.) (c) For what value of $x$ does $f(x)=g(x)$ ? $x=1$ (Type an integer or a simplified fraction.) (d) For which values of $x$ is $f(x) \leq g(x)$ ? $\square$ (Type your answer in interval notation. Use integers or fractions for any numbers in the expression.)

Studdy Solution

STEP 1

1. We have 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 provide solutions in integer or simplified fraction form and use interval notation where applicable.

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

To solve $f(x) = 0$ , set the function equal to zero and solve for $x$ :
$6x - 1 = 0$
Add 1 to both sides:
$6x = 1$
Divide both sides by 6:
$x = \frac{1}{6}$

STEP 4

To solve $f(x) > 0$ , set the inequality and solve for $x$ :
$6x - 1 > 0$
Add 1 to both sides:
$6x > 1$
Divide both sides by 6:
$x > \frac{1}{6}$
In interval notation, the solution is:
$\left(\frac{1}{6}, \infty\right)$

STEP 5

To solve $f(x) = g(x)$ , set the two functions equal and solve for $x$ :
$6x - 1 = -3x + 8$
Add $3x$ to both sides:
$9x - 1 = 8$
Add 1 to both sides:
$9x = 9$
Divide both sides by 9:
$x = 1$

STEP 6

To 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:
$x \leq 1$
In interval notation, the solution is:
$\left(-\infty, 1\right]$

STEP 7

Graph $y = f(x)$ and $y = g(x)$ to find the intersection point.
The intersection point of the lines $y = 6x - 1$ and $y = -3x + 8$ is at $x = 1$ .
The corresponding $y$ -value is:
$y = 6(1) - 1 = 5$
So, the intersection point is $(1, 5)$ .

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Studdy Solution

STEP 1

1. We have 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 provide solutions in integer or simplified fraction form and use interval notation where applicable.

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

To solve $f(x) = 0$ , set the function equal to zero and solve for $x$ :
$6x - 1 = 0$
Add 1 to both sides:
$6x = 1$
Divide both sides by 6:
$x = \frac{1}{6}$

STEP 4

To solve $f(x) > 0$ , set the inequality and solve for $x$ :
$6x - 1 > 0$
Add 1 to both sides:
$6x > 1$
Divide both sides by 6:
$x > \frac{1}{6}$
In interval notation, the solution is:
$\left(\frac{1}{6}, \infty\right)$

STEP 5

To solve $f(x) = g(x)$ , set the two functions equal and solve for $x$ :
$6x - 1 = -3x + 8$
Add $3x$ to both sides:
$9x - 1 = 8$
Add 1 to both sides:
$9x = 9$
Divide both sides by 9:
$x = 1$

STEP 6

To 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:
$x \leq 1$
In interval notation, the solution is:
$\left(-\infty, 1\right]$

STEP 7

Graph $y = f(x)$ and $y = g(x)$ to find the intersection point.
The intersection point of the lines $y = 6x - 1$ and $y = -3x + 8$ is at $x = 1$ .
The corresponding $y$ -value is:
$y = 6(1) - 1 = 5$
So, the intersection point is $(1, 5)$ .

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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

To solve f(x)=0 f(x) = 0 f(x)=0, set the function equal to zero and solve for x x x: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

To solve f(x)>0 f(x) > 0 f(x)>0, set the inequality and solve for x x x: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​In interval notation, the solution is:(16,∞) \left(\frac{1}{6}, \infty\right) (61​,∞)

STEP 5

To solve f(x)=g(x) f(x) = g(x) f(x)=g(x), set the two functions equal and solve for x x x: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

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.

To solve $f(x) = 0$ , set the function equal to zero and solve for $x$ :
$6x - 1 = 0$
Add 1 to both sides:
$6x = 1$
Divide both sides by 6:
$x = \frac{1}{6}$

To solve $f(x) > 0$ , set the inequality and solve for $x$ :
$6x - 1 > 0$
Add 1 to both sides:
$6x > 1$
Divide both sides by 6:
$x > \frac{1}{6}$
In interval notation, the solution is:
$\left(\frac{1}{6}, \infty\right)$

To solve $f(x) = g(x)$ , set the two functions equal and solve for $x$ :
$6x - 1 = -3x + 8$
Add $3x$ to both sides:
$9x - 1 = 8$
Add 1 to both sides:
$9x = 9$
Divide both sides by 9:
$x = 1$