Modulus Revision Notes for Leaving Cert Mathematics

Modulus Inequalities

Modulus inequalities involve the absolute value of a linear function of a variable in the form $|ax+b|$ .

Example

infoNote

Solve for $x$ in the following inequality :

|2x-3|<5

Square both sides :

We can rewrite the inequality as :

\begin{align*} (2x-3)^2 &< 5^2 \\\\ 4x^2-12x+9 &< 25 \\\\ 4x^2-12x-16 &< 0 \\\\ (x-4)(x+1)&< 0 \end{align*}

Same as before, let's do the region test, pick two values outside $-1$ and $4$ and a value between.

$x=0$

\begin{align*} |2(0)-3|&<5 \\ |-3|&<5 \\ 3 &< 5 \end{align*}

$x=5$

\begin{align*} |2(5)-3|&<5 \\ |10-3|&<5 \\ 7 &\not< 5 \end{align*}

$x=-2$

\begin{align*} |2(-2)-3|&<5 \\ |-4-3|&<5 \\ 7 &\not< 5 \end{align*}

-1<x<4