Modulus Equations & Inequalities (DP IB Analysis & Approaches (AA): HL): Revision Note

Modulus equations

How do I solve modulus equations graphically?

• To solve $\left|f\left(x\right)\right|=g\left(x\right)$ (or $\left|f\left(x\right)\right|=\left|g\left(x\right)\right|$) graphically

  • Draw $y=\left|f\left(x\right)\right|$ and $y=g\left(x\right)$ (or $y=\left|g\left(x\right)\right|$) into your GDC

    • Find the $x$-coordinates of the points of intersection

How do I solve modulus equations using algebra?

• To solve either $\left|f\left(x\right)\right|=g\left(x\right)$ or $\left|f\left(x\right)\right|=\left|g\left(x\right)\right|$ using algebra, the process is the same:

  • split into two equations

    • $f\left(x\right)=g\left(x\right)$

    • $f\left(x\right)=-g\left(x\right)$

  • Solve both equations

  • Check that the solutions work in the original equation

• e.g. $|x-2|=2x-3$ splits into $x-2=2x-3$ and $x-2=-\left(2x-3\right)$

  • the first equation gives $x=1$

  • but $x=1$ is not a solution to $|x-2|=2x-3$

    • as $\left|\right(1)-2|=1$ and $2\left(1\right)-3=-1$

  • the second equation gives $x=\frac{5}{3}$

    • which does satisfy $|x-2|=2x-3$ so is the solution

• A sketch can help, even when solving algebraically

Modulus inequalities

How do I solve modulus inequalities?

• To solve modulus inequalities

  • first solve the modulus equation

    • by replacing the inequality sign with =

  • then use a graphical method to find the intervals of $x$ that satisfy the inequality

• To solve $G_1\le G_2$, $G_1\ge G_2$, $G_1<G_2$ or $G_1>G_2$ where

  • $G_1$ is graph 1

  • $G_2$ is graph 2

  • STEP 1
    Sketch $G_1$ and $G_2$

  • STEP 2
    Locate the $x$-coordinates of the points of intersection

    • these would be $x$-axis intercepts if $G_2=0$

  • STEP 3
    Determine which part(s) of the graph(s) satisfy the inequality

    • $G_1\le G_2$ or $G_1<G_2$are where graph 1 is below than graph 2

    • $G_1\ge G_2$ or $G_1>G_2$ are where graph 1 is above than graph 2

  • STEP 4
    Write the range of values of $x$ for these regions

    • using strict inequalities if $G_1<G_2$ or $G_1>G_2$

    • or 'equal to' inequalities if $G_1\le G_2$ or $G_1\ge G_2$

• An alternative method is to use a sign table

  • e.g. where you substitute a numerical value from each of the possible intervals of $x$ into the original inequality

    • The solutions are the regions for which the original inequality is true