QuestionSolve the equation: $\left|\frac{5}{x-3}\right|=15$

Studdy Solution

STEP 1

Assumptions1. The absolute value of a number is its distance from zero on the number line, so it is always non-negative. . The absolute value of a number can be either positive or zero, but never negative.
3. The equation is $\left|\frac{5}{x-3}\right|=15$ .

STEP 2

We can start by removing the absolute value. This gives us two equations because the absolute value of a number can be either the number itself if it's positive or zero, or the negative of the number if it's negative.
$\frac{5}{x-}=15 \quad and \quad \frac{5}{x-}=-15$

STEP 3

Now, we solve each equation separately. Let's start with the first equation. We can multiply both sides by $(x-3)$ to get rid of the denominator on the left side.
$(x-3)\times\frac{5}{x-3}=15\times(x-3)$

STEP 4

implify the equation.
$=15x-45$

STEP 5

Rearrange the equation to isolate $x$ .
$15x=5+45$

STEP 6

implify the right side of the equation.
$15x=50$

STEP 7

Divide both sides by15 to solve for $x$ .
$x=\frac{50}{15}$

STEP 8

implify the fraction to get the value of $x$ .
$x=\frac{10}{3}$

STEP 9

Now, let's solve the second equation. Again, we can multiply both sides by $(x-3)$ to get rid of the denominator on the left side.
$(x-3)\times\frac{5}{x-3}=-15\times(x-3)$

STEP 10

implify the equation.
$5=-15x+45$

STEP 11

Rearrange the equation to isolate $x$ .
$15x=45-5$

STEP 12

implify the right side of the equation.
$15x=40$

STEP 13

Divide both sides by15 to solve for $x$ .
$x=\frac{40}{15}$

STEP 14

implify the fraction to get the value of $x$ .
$x=\frac{8}{3}$ So, the solutions to the equation $\left|\frac{}{x-3}\right|=$ are $x=\frac{10}{3}$ and $x=\frac{8}{3}$ .

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