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.

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    $\begingroup$ When you divide by a negative number, the sign needs to reverse as well. $\endgroup$ – Ininterrompue Jan 11 at 14:16
  • $\begingroup$ I did reverse it. -14 became 7 $\endgroup$ – Henry Cooper Jan 11 at 14:16
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    $\begingroup$ Sorry. I meant the inequality sign goes from > to <. $\endgroup$ – Ininterrompue Jan 11 at 14:17
  • $\begingroup$ You didn’t reverse the inequality sign. $\endgroup$ – KM101 Jan 11 at 14:17
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    $\begingroup$ 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! $\endgroup$ – 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:

$$-10 < -4$$

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    $\begingroup$ The glorious feeling of catharsis! Thanks Tony, I'll accept your answer in 7 minutes :) $\endgroup$ – Henry Cooper Jan 11 at 14:22
  • $\begingroup$ Can't help but think the example would have been clear with divide by -1 but good answer $\endgroup$ – 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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  • 1
    $\begingroup$ Did you lose your 2 somewhere in there? Or in fact gain an unwanted 2 I think... $\endgroup$ – Chris Jan 11 at 14:47
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    $\begingroup$ He meant $-14$ in the first inequality. $\endgroup$ – KM101 Jan 11 at 15:17

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