Answer: -11 < x < -4
x is some number between -11 and -4; x cannot equal -11, x cannot equal -4.
the graph on the number line will show two open circles at -11 and -4 with shading between the two open circles (the open circles are not filled in).
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Work Shown on how I got that answer:
2 - |(2/5)*x + 3| > 3/5
2 - |(2/5)*x + 3| - 2 > 3/5 - 2 ... see note 1 (below)
-|(2/5)*x + 3| > 3/5 - 10/5 ... see note 2
-|(2/5)*x + 3| > -7/5
|(2/5)*x + 3| < 7/5 ... see note 3
-7/5 < (2/5)*x + 3 < 7/5 ... see note 4
5*(-7/5) < 5*( (2/5)*x + 3 ) < 5*(7/5) ... see note 5
-7 < 5*(2/5)*x + 5*3 < 7
-7 < 2x + 15 < 7
-7-15 < 2x + 15-15 < 7-15 ... see note 6
-22 < 2x < -8
-22/2 < 2x/2 < -8/2 ... see note 7
-11 < x < -4
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notes:
- note 1: I subtracted 2 from both sides
- note 2: We can write "2" as "10/5" since 10/5 = 2, this helps us combine the fractions on the next step.
- note 3: I multiplied both sides by -1. This will flip the inequality sign.
- note 4: Use the rule that if |x| < k, then -k < x < k for some positive number k.
- note 5: Multiply all three sides by 5 so the denominators of 5 cancel out (ie the fractions go away).
- note 6: Subtract 15 from all three sides to undo the +15 in the middle.
- note 7: Divide all three sides by 2 to undo the "multiplication of 2" done on the x. This fully isolates x.
