# Thread: Modulus of function help

1. ## Modulus of function help

Hello can anyone please tell me the steps in solving a modulus of a function?
|2x +3| =5

2. ## Re: Modulus of function help

By definition, either $\displaystyle 2x+3=5$ or $\displaystyle -(2x+3)=5$. Alternatively, you could square both sides, to give $\displaystyle (2x+3)^2=25$ and solve as a quadratic.

3. ## Re: Modulus of function help

So if i get a question like this for exam all i have to do is square both sides???? AND THANKS ALOT>>>>>

4. ## Re: Modulus of function help

It depends. Squaring both sides won't work for $\displaystyle |2x+3|>5$, for example. If you have an equation, it's a valid approach. I'd suggest knowing both methods, so that you can verify your answer.

5. ## Re: Modulus of function help

Originally Posted by mathkid12
Hello can anyone please tell me the steps in solving a modulus of a function?
|2x +3| =5

You need to know the basic definitions of the modulus function, namely

\displaystyle \displaystyle \begin{align*} |X| = \begin{cases} \phantom{-} X \textrm{ if }X \geq 0 \\ -X \textrm{ if } X < 0 \end{cases} \end{align*}

and \displaystyle \displaystyle \begin{align*} \sqrt{X^2} = |X| \end{align*}.

Also, by some geometric reasoning, you should know that \displaystyle \displaystyle \begin{align*} |X - c| = a \end{align*} means "\displaystyle \displaystyle \begin{align*} X \end{align*} is \displaystyle \displaystyle \begin{align*} a \end{align*} units away from \displaystyle \displaystyle \begin{align*} c \end{align*}."

So in the case of your equation \displaystyle \displaystyle \begin{align*} |2x + 3| = 5 \end{align*}, by the definition of the modulus function, that means

\displaystyle \displaystyle \begin{align*} |2x + 3| &= \begin{cases} \phantom{-(} 2x + 3 \phantom{)} \textrm{ if }2x + 3 \geq 0 \\ -(2x + 3) \textrm{ if } 2x + 3 < 0 \end{cases} \\ &= \begin{cases} \phantom{-} 2x + 3 \textrm{ if } x \geq -\frac{3}{2} \\ -2x - 3 \textrm{ if }x < -\frac{3}{2} \end{cases} \end{align*}

So now you need to solve the equation for the two separate cases.

Case 1: \displaystyle \displaystyle \begin{align*} x \geq -\frac{3}{2} \end{align*}

\displaystyle \displaystyle \begin{align*} |2x + 3| &= 5 \\ 2x + 3 &= 5 \\ 2x &= 2 \\ x &= 1 \end{align*}

This solution is valid since it fits in with the restriction.

Case 2: \displaystyle \displaystyle \begin{align*} x < -\frac{3}{2} \end{align*}

\displaystyle \displaystyle \begin{align*} |2x + 3| &= 5 \\ -2x - 3 &= 5 \\ -2x &= 8 \\ x &= -4 \end{align*}

This solution is also valid since it fits with the restriction.

Now if you wanted to use the geometric interpretation...

\displaystyle \displaystyle \begin{align*} |2x + 3| &= 5 \\ 2\left|x + \frac{3}{2}\right| &= 5 \\ \left|x + \frac{3}{2}\right| &= \frac{5}{2} \\ \left|x - \left(-\frac{3}{2}\right)\right| &= \frac{5}{2} \end{align*}

So you are looking for the values of \displaystyle \displaystyle \begin{align*} x \end{align*} which are \displaystyle \displaystyle \begin{align*} \frac{5}{2} \end{align*} units away from \displaystyle \displaystyle \begin{align*} -\frac{3}{2} \end{align*}.

Clearly these are \displaystyle \displaystyle \begin{align*} -\frac{3}{2} - \frac{5}{2} = -4 \end{align*} and \displaystyle \displaystyle \begin{align*} -\frac{3}{2} + \frac{5}{2} = 1 \end{align*}.

6. ## Re: Modulus of function help

wow thanks THANKS alot your help really appreciated