SOLUTION: |4a+3|is less than or equal to 5

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Question 173700: |4a+3|is less than or equal to 5
Found 2 solutions by solver91311, gonzo:
Answer by solver91311(24713) About Me  (Show Source):
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abs%284a%2B3%29%3C=5 means that 4a%2B3%3C=5 and -%284a%2B3%29%3C=5, so:

4a%2B3%3C=5 → 4a%3C=2 → a%3C=2%2F4 → a%3C=1%2F2

and

-%284a%2B3%29%3C=5 → -4a-3%3C=5 → -4a%3C=8 → a%3E=8%2F%28-4%29 (Note inequality sense change because of dividing by a negative number) → a%3E=-2

Therefore a is any real number in the interval -2%3C=a%3C=1%2F2

Answer by gonzo(654) About Me  (Show Source):
You can put this solution on YOUR website!
|4a+3| <= 5
divide into 2 equations:
positive side:
4a+3 <=5
and
negative side:
4a+3 >= -5
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4a+3<=5 becomes:
4a<=2
a<=.5
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4a+3>=-5 becomes:
4a>=-8
a>=-2
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you have:
a<=.5
a>=-2
rewritten, this is:
-2<=a<=.5
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to prove:
let a = -2
|4a+3| = |-8+3| = |-5| = 5 <= 5
let a = .5
|4a+3| = |2+3| = 5 <= 5
we're good on the limits.
take one within the limits
let a = 0
|4a+3| = |3| = 3 <= 5 (ok)
take one below the limits.
let a = -3
|4a+3| = |-12+3| = |-9| = 9 NOT <= 5 (not ok)
take one above the limits.
let a = 1
|4a+3| = |4+3| = |7| = 7 NOT <= 5 (not ok).
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looks like the values of:
-2<=a<=.5
are good.
that's your answer.
check this website out for a fuller explanation.
http://www.purplemath.com/modules/absineq.htm