# Getting wrong answer for absolute value inequality and not sure why

The question:

The function p is defined by $$-2|x+4|+10$$. Solve the equality $$p(x) > -4$$

Here were my steps to solving this:

1.) Subtract 10 from both sides -> $$-2|x+4| > -14$$

2.) Divide both sides by -2 -> $$|x+4|>7$$

3.) $$x+4$$ should therefore be 7 units or greater from zero on the number line, meaning either greater than 7 or less than -7:

$$x+4 > 7$$

$$x+4 < -7$$

4.) Subtract 4 from both sides:

$$x > 3$$

$$x < -11$$

Graphing this, I see my signs are the wrong way round but I'm not quite sure where I've gone wrong.

• When you divide by a negative number, the sign needs to reverse as well. – Ininterrompue Jan 11 at 14:16
• I did reverse it. -14 became 7 – Henry Cooper Jan 11 at 14:16
• Sorry. I meant the inequality sign goes from > to <. – Ininterrompue Jan 11 at 14:17
• You didn’t reverse the inequality sign. – KM101 Jan 11 at 14:17
• A little tip: when you know the answer is wrong, you can find faulty steps by substituting a bad value into the previous steps. You know that the answer shouldn't include $x > 3$ as your working shows, so try substituting $x = 4$ into each step. Initially, you get a false inequality, but at the first step of bad working, it magically becomes true! – Theo Bendit Jan 11 at 14:33

$$20$$ is greater than $$8$$, right?

$$20 > 8$$

Now divide both sides by $$-2$$:

$$-10 > -4$$

Whoops! That's not right. This is because when you multiply or divide an inequality by a negative number, you must change the sense of the inequality: $$>$$ becomes $$<$$, and $$\le$$ becomes $$\ge$$ etc:

$$-10 < -4$$

• The glorious feeling of catharsis! Thanks Tony, I'll accept your answer in 7 minutes :) – Henry Cooper Jan 11 at 14:22
• Can't help but think the example would have been clear with divide by -1 but good answer – ArtB Jan 12 at 22:33

$$-2|x+4|>-14$$ implies that $$|x+4|<7$$ (the greater than becomes smaller than because you multiply by a negative number).

The rest is all good.

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• Did you lose your 2 somewhere in there? Or in fact gain an unwanted 2 I think... – Chris Jan 11 at 14:47
• He meant $-14$ in the first inequality. – KM101 Jan 11 at 15:17