Solve the inequality |x−5| < 3|2x+2|

Full worked solution

1. Eliminate absolute value signs by squaring

   Since all absolute value expressions are non-negative for all real $x$, we can square both sides of the inequality without changing its direction, using the identity $|a{|}^2=a^2$ for all real $a$.

   $$(x - 5)^2 < \left(3|2x + 2|\right)^2$$

   Simplify the right-hand side, expand both sides, and rearrange all terms to one side of the inequality:

   $$(x - 5)^2 < 9(2x + 2)^2 \ x^2 - 10x + 25 < 9(4x^2 + 8x + 4) \ x^2 - 10x + 25 < 36x^2 + 72x + 36 \ 0 < 35x^2 + 82x + 11$$

   We now have the standard quadratic inequality $35x^2+82x+11>0$.

   Note.

   Squaring both sides is only safe when both sides are non-negative, which is always true for absolute value expressions. Never square an inequality with unknown signs on either side.

2. Find critical values by solving the quadratic equation

   First solve the corresponding quadratic equation $35x^2+82x+11=0$ using the quadratic formula $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$, where $a=35$, $b=82$, and $c=11$. First calculate the discriminant:

   $$\Delta = b^2 - 4ac = 82^2 - 4(35)(11) = 6724 - 1540 = 5184 = 72^2$$

   Now compute the roots:

   $$x = \frac{-82 \pm 72}{2(35)} = \frac{-82 \pm 72}{70}$$

   First root: $x=\frac{-82+72}{70}=\frac{-10}{70}=-\frac{1}{7}$ Second root: $x=\frac{-82-72}{70}=\frac{-154}{70}=-\frac{11}{5}$ The critical values are $x=-\frac{11}{5}$ (or $-2.2$) and $x=-\frac{1}{7}$ (or approximately $-0.143$).

   Note.

   The quadratic formula works for all quadratic equations, so it is a reliable method to find critical points even when factoring is not obvious.

3. Determine the solution interval from the quadratic graph

   The quadratic function $y=35x^2+82x+11$ has a positive leading coefficient, so its graph is an upward-opening parabola that crosses the $x$-axis at the critical points $x=-\frac{11}{5}$ and $x=-\frac{1}{7}$. An upward-opening parabola is above the $x$-axis (i.e. $y>0$) outside the interval between its two roots.

What this tests