$How do you do absolute value equations with inequalities?$

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$How do you do absolute value equations with inequalities?$

So if you have an absolute value equation like $\left | 5+x \right | >7-2x$ how do you go about solving it.

I know how to solve absolute value equations with an equals sign but not with inequalities.

Thanks in advance.

Algebra

Math Gnome

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When dealing with absolute values you need to consider two cases (I) The expression inside the absolute value is positive (II) The expression inside the absolute value is negative.

In this particular example the expression inside the absolute value is 5+x:

Case I. The case $5+x\geq 0$ or $x\geq -5$: Then |5+x|= 5+x. So the inequality becomes
\[5+x >7-2x \Rightarrow 3x>2 \Rightarrow x>\frac{2}{3}.\]
Since we are in case I, both conditions
\[x\geq -5 \text{and} x>\frac{2}{3} \]
must be true. They are both true if $x>\frac{2}{3}$ or in the interval $(\frac{2}{3}, \infty)$.

Case II. The case $5+x< 0$ or $x<-5$: Then $|5+x|= -(5+x)=-5-x$. So the inequality becomes
\[-5-x >7-2x \Rightarrow x>12.\]
Since we are in case II, both conditions
\[x< -5 \text{and} x>12 \]
which is impossible. Hence case II does not lead to any solution.

So the given inequality holds if and only if
\[\bf{x>\frac{2}{3}}.\]

Erdos

4.8K

Math Gnome

0

Where did you the get 5+x ≥ 0 or x ≥ -5 part from?
- Erdos
  
  +1
  
  You add -5 to both sides of the inequality.
Erdos

+1

You can do pretty much everything with inequalities that you do with equalities. The only difference is that whenever you multiply or divide both sides by a negative number, the direction of the inequality switches.
Math Gnome

0

Thanks for the reply what I meant is why do you put a zero on the other side of the inequality? Where does the zero come from.
- Erdos
  
  +1
  
  This comes from the basic fact about absolute values: (i) If A >= 0 then |A|=A (ii) If A<0, then |A|=-A. This is why in general we consider two cases A>=0 and A<0. In this example A=5+x.

$How do you do absolute value equations with inequalities?$

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