Equations with Two Moduli | CAIE 9709 Pure Mathematics 2 Notes

Pure Mathematics 2

Mathematics9709

Equal size does not require equal sign

$|u|=|v|$ means that $u$ and $v$ have equal size: they are equal or opposites. Both sides are already non-negative, so there is no extra sign restriction on $v$ .

\begin{gathered}|u|=|v|\\\Updownarrow\\u=v\ \text{or}\ u=-v\end{gathered}

The double arrow means the statements are equivalent: each implies the other.

Worked example

Solve |3x + 4| = |x + 5|.

\begin{gathered}3x+4=x+5\\2x=1,\quad x=\tfrac12\\[4pt]3x+4=-(x+5)\\4x=-9,\quad x=-\tfrac94\end{gathered}

At $1/2$ both insides are $11/2$ . At $-9/4$ they are $-11/4$ and $11/4$ . Both roots work.

Squaring is another exact route

Equal non-negative numbers have equal squares, and vice versa. Since $|u|^2=u^2$ , we can also write $u^2=v^2$ . Use the difference of two squares to avoid unnecessary expansion:

u^2-v^2=(u-v)(u+v)=0

A product is zero only if at least one factor is zero. These factors give exactly the same two cases as before. For linear expressions, the two linear equations are usually shorter.

Your turnPractice

Solve $|2x-5|=|x+1|$ . Then decide how many solutions $|x-2|=|2-x|$ has.

Show worked answer

\begin{gathered}2x-5=x+1\\\text{or}\\2x-5=-(x+1)\end{gathered}

Answer

x=6\quad\text{or}\quad x=4/3

The second equation is true for every real $x$ , because $2-x=-(x-2)$ . Two algebraic cases do not guarantee two distinct solutions.