Absolute Value Inequalities Analytical Approach

For $$|x-3|-|2x+1|<0$$, I considered adding $$|2x+1|$$ on both sides and solving it. $$|x-3|<|2x+1|$$ $$x-3< |2x+1|$$ $$x-3 < 2x+1$$ However, I keep getting $$x>-4$$ and $$x<2/3$$ instead of $$x<-4$$ and $$x>2/3$$. I know I can do a graphical approach and find the values, but why do I keep getting the wrong answer?

• How do you justify simply erasing the absolute-value signs? – Henning Makholm Dec 19 '18 at 1:19

Analytically you may also proceed as follows:

$$\begin{eqnarray*} |x-3| < |2x+1| & \Leftrightarrow & (x-3)^2 < (2x+1)^2 \\ & \Leftrightarrow & 0 < (2x+1)^2 -(x-3)^2 \stackrel{a^2-b^2 =(a-b)(a+b)}{=} (2x+1 - (x-3))(2x+1 + (x-3))\\ & \Leftrightarrow & 0 < (x+4)(3x-2) \\ & \Leftrightarrow & \boxed{x< -4 \mbox{ or } x>\frac{2}{3}}\\ \end{eqnarray*}$$

You have $$|x-3| < |2x+1|$$. Now there are 4 possible cases:

1) $$x-3\ge 0$$ and $$2x + 1 \ge 0$$.

This means:

a) $$x - 3 \ge 0$$ so $$x \ge 3$$.

b) $$2x + 1 \ge 0$$ so $$x \ge -\frac 12$$

c) $$x - 3 < 2x + 1$$ so $$-4 < x$$ or $$x > -4$$.

Combining we get $$x \ge 3$$.

2) $$x-3 < 0$$ and $$2x +1 \ge 0$$.

This means

a) $$x - 3 < 0$$ so $$x < 3$$

b) $$2x + 1 \ge 0$$ so $$x \ge -\frac 12$$

c) $$3-x < 2x + 1$$ so $$2 < 3x$$ and $$x > \frac 23$$

Combining we get $$\frac 23 < x < 3$$.

Case 3) $$x-3 \ge 0$$ and $$2x + 1 < 0$$ then

a) $$x -3 \ge 0$$ and $$x > 3$$

b) $$2x + 1 < 0$$ and $$x < -\frac 12$$

c) $$x-3 < -1-2x$$ and $$3x < 2$$ and $$x < \frac 23$$.

Combining we get contradictions. This is not possible.

Case 4) $$x-3 < 0$$ and $$2x + 1 < 0$$ then

a) $$x - 3 < 0$$ so x < 3$b) $$2x +1< 0$$ so x < -\frac 12$.

c) $$3-x < -2x - 1$$ so $$x < -4$$

Combining we get $$x < -4$$.

So we have 3 possible results $$x \ge 3$$ or $$\frac 23 < x < 3$$ or $$x < -4$$. Combining we get either $$x < -4$$ of $$x > \frac 23$$.

• If $3 - x < 2x + 1$, then $2 < 3x \implies x > \frac{2}{3}$. – N. F. Taussig Dec 19 '18 at 10:24
• Dang. Arithmetic in your head is hard ..... – fleablood Dec 19 '18 at 17:18
• Yee gods! I made at least 3 errors and a totally wrong answer! Still, I think cases are the most straight forward way to learn it. – fleablood Dec 19 '18 at 17:43