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 September 24th, 2013, 08:13 AM #1 Newbie   Joined: Jul 2013 Posts: 21 Thanks: 0 Absolute Value Inequality with Absolute Values on Both Sides How do I solve these kinds of inequalities??? My professor did not teach it and I couldn't find much information on the Internet. And we're going to have a test about them tomorrow!! Example: $\left | 7-3x \right |\geq \left | 3x+8 \right |\$ Some say I should square both sides. Some say I should separate them. Some say I should let the greater/less than sign to be an equal sign first. I'm so confused. Please enlighten me. Thank you!
 September 24th, 2013, 09:27 AM #2 Global Moderator     Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,975 Thanks: 1156 Math Focus: Elementary mathematics and beyond Re: Absolute Value Inequality with Absolute Values on Both S 1. $7\,-\,3x\,=\,8\,+\,3x\,\Rightarrow\,x\,=\,-\frac16 \\ |7\,-\,3x|\,\ge\,|8\,+\,3x|\,\Rightarrow\,x\,\le\,-\frac16 \\$ 2. $|7\,-\,3x|^2\,\ge\,|8\,+\,3x|^2 \\ 49\,-\,42x\,+\,9x^2\,\ge\,64\,+\,48x\,+\,9x^2 \\ -90x\,\ge\,15 \\ x\,\le\,-\frac16$
September 24th, 2013, 09:36 AM   #3
Math Team

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Re: Absolute Value Inequality with Absolute Values on Both S

Hello, shiseonji!

Quote:
 $|7\,-\,3x| \:\ge\: |3x\,+\,8|$ Some say I should square both sides. Some say I should separate them. Some say I should let the greater/less than sign to be an equal sign first. [color=blue]These are all good ideas.[/color]

I would use the last approach . . .
$\;\;\;7\,-3x\:=\:3x\,+\,8 \;\;\;\Rightarrow\;\;\;-6x \:=\:1 \;\;\;\Rightarrow\;\;\; x \:=\:-\frac{1}{6}$

This divides the number line into two intervals:

$\;\;\;\begin{array}{ccc}---- & \bullet & ---- \\ \\ & -\frac{1}{6}\; \end{array}$

Test a value of $x$ in each interval.

$\begin{array}{cccccccccc}x = -1: && |7\,-\,(-1)| \:\ge\:|3(-1)\,+\,8| &&\Rightarrow&& |8| \,\ge\,|5|& \text{ . . . true} \\ \\ \\
x = 0: && |7\,-\,0| \:\ge\:|0\,+\,8| &&\Rightarrow&& |7|\:\ge\:|8| & \text{ . . . false} \end{array}$

Therefore:[color=beige] .[/color]$x \:\le\:-\frac{1}{6}$

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# absolute value on both sides of inequality

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