Question Find $x$ where $f(x)=6x-5=0$ . Graph $y=f(x)$ and $y=g(x)=-3x+4$ , and find the point where they intersect.

Studdy Solution

STEP 1

Assumptions1. The function $f(x)$ is defined as $f(x)=6x-5$ . The function $g(x)$ is defined as $g(x)=-3x+4$
(a) Solve $f(x)=0$

STEP 2

To solve the equation $f(x)=0$ , we need to set $f(x)$ equal to zero and solve for $x$ . $f(x) =6x -5 =0$

STEP 3

Now, solve the equation for $x$ . $6x -5 =0$

STEP 4

Add to both sides of the equation to isolate $6x$ on one side.
$6x =$

STEP 5

Divide both sides of the equation by to solve for $x$ .
$x = \frac{5}{}$ (b) Solve $f(x)>0$

STEP 6

To solve the inequality $f(x)>0$ , we need to set $f(x)$ greater than zero and solve for $x$ . $f(x) =6x -5 >0$

STEP 7

Now, solve the inequality for $x$ .
$6x -5 >0$

STEP 8

Add5 to both sides of the inequality to isolate $6x$ on one side.
$6x >5$

STEP 9

Divide both sides of the inequality by6 to solve for $x$ .
$x > \frac{5}{6}$ (c) Solve $f(x)=g(x)$

STEP 10

To solve the equation $f(x)=g(x)$ , we need to set $f(x)$ equal to $g(x)$ and solve for $x$ .
$6x -5 = -3x +4$

STEP 11

Add $3x$ to both sides of the equation to isolate $x$ terms on one side.
$9x -5 =4$

STEP 12

Add5 to both sides of the equation to isolate $9x$ on one side.
$9x =9$

STEP 13

Divide both sides of the equation by9 to solve for $x$ .
$x =$ (d) Solve $f(x) \leq g(x)$

STEP 14

To solve the inequality $f(x)\leq g(x)$ , we need to set $f(x)$ less than or equal to $g(x)$ and solve for $x$ .
$6x - \leq -3x +4$

STEP 15

Add $3x$ to both sides of the inequality to isolate $x$ terms on one side.
$9x -5 \leq4$

STEP 16

Add5 to both sides of the inequality to isolate $9x$ on one side.
$9x \leq9$

STEP 17

Divide both sides of the inequality by9 to solve for $x$ .
$x \leq$ (e) Graph $y=f(x)$ and $y=g(x)$ and find the point that represents the solution to the equation $f(x)=g(x)$ .

STEP 18

To graph the functions $f(x)$ and $g(x)$ , plot the lines $y=6x-5$ and $y=-3x+4$ on the same graph.

STEP 19

The point of intersection of the two lines represents the solution to the equation $f(x)=g(x)$ . We have already calculated this point in part (c) to be $x=1$ . Substitute $x=1$ into either $f(x)$ or $g(x)$ to find the $y$ -coordinate of the point of intersection.
$y = f(1) =6(1) -5 =1$ So, the point of intersection is $(1,1)$ .

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