Question

the abs. value of (2x-7) is less than or equal to 1

Answer 1

So depending on which way the inequality is facing, you set up the problem differently. If we have something like
\[|x| \le a\], then we set up our problem like this:
\[-a \le x \le a\]
Whatever is inside of the absolute value goes inbetween the negative of the number and the positive of the number. SO basically, you would set up your specific problem liek this:

\[-1 \le 2x-7 \le 1\]So now the goal is to solvefor x. You get x by itself the same way you would in an algebra problem liek 2x + 4 = 8, except this time if you subtract, add, divide, multiply, you must do it to every single portion of the inequality. So to get the x by itself, we would first add 7 to all 3 portions of the compound inequality. This gives us:
\[6 \le 2x \le 8\]Finally just divide everything by 2 to get this as your answer:

\[3 \le x \le 4\]

Kinda make sense?

Answer 2

Thank you so much! I appreciate your help,
To divide with absolute value, would you multiply both sides by the denominator, so to cancel the denominator?

Answer 3

If you have something like x/2 in between the inequalities then yep, youd multiply everythign by 2 to cancel the denominator.

Answer 4

If I had abs. value of ( 3x-1/-4) is less than two, I would multiply both sides by -4?

Answer 5

\[|\frac{ 3x-1 }{ -4 }| < 2\] correct?

Answer 6

yes

Answer 7

Alright, so we do this then:

\[-2 < \frac{ 3x-1 }{ -4 }< 2\]
Its at this point that you would multiply all 3 portions by -4 to get:

\[8 > 3x-1 > -8\]Just remember that when you multiply or divide by a negative that you must flip the inequalities.

Answer 8

in the case of the abs value (x/x-1) equals two...would you also do the neg and pos sides then multiply by x-1?

Answer 9

Right, you would have two equations since it is an equal sign.

x/x-1 = 2 and
x/x-1 = -2

Then multiply both sides by x-1 in each equation and solve for x in both.

Answer 10

okay, great. thank you!!

Answer 11

Yep yep ^_^

Answer 12

would I multiply by x-1/x-1 or x-1/1?

Answer 13

For the last situation you mentioned?

Answer 14

yes

Answer 15

\[\frac{ x }{ x-1 }=2\implies x = 2(x-1)\]Just multiplied both sides by x-1

Answer 16

x equals 1?

Answer 17

x = 2x - 2
+2 +2

2 + x = 2x
-x -x

2 = x

Now you just do the same thing for the negative version.

Answer 18

thank you!

Answer 19

Sure : )