3|x−4|+4>14
PHASE 1. Isolate the absolute value expression.
Subtract from both sides.
3|x−4| >14 -4=10
3|x−4| > 10
Divide both sides by 3.
|x−4| > 10/3
Leave the improper fraction as it is for now. We'll convert it to a proper fraction at the end.
Phase 1 is complete now. The absolute value expression is on one side of the inequality, and everything else is on the other side. This process isbknown as isolating the absolute value expression.
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PHASE 2. Divide and conquer.
Either x-4 ≥ 0 or x-4<0.
CASE 1. x-4 ≥ 0.
In this case, |x-4| = x-4, so we cam drop the bars and solve for x.
x-4 > 10/3
Add 4 to both sides:
x > 10/3 + 4
Convert the 4 to an equivalent number of thirds:
x > 10/3 + 4*3/3 = 10/3 + 12/3 = 22/ 3 = 7 1/3
x > 7 1/3
Now that we are finished with this case, we convert to proper mixed fraction.
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CASE 2: x-4 < 0
In this case, |x-4| = -(x-4). Replace the absolute value expression with this expression and solve for x.
-(x-4) > 10/3
Multiply both sides by -1. Recall that we must reverse the direction of the inequality sign when we multiply or divide by negative numbers.
x-4 < -10/3
Add 4 to both sides:
x < -10/3 + 4
As before, convert the 4 to a multiple of thirds.
x < -10/3 + 4×3/3 = -10/3 + 12/3 = 2/3
x < 2/3
END of CASE 2.
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PHASE 3. WRAP UP AND CHECK.
Combine the results of the two cases.
x < 2/3 OR x > 7 1/3
A quick check will involve numbers in each of three areas:
x< 2/3
2/3 ≤ x ≤ 7 1/3
and
x < 7 1/3
Try 0:
3|x-4| +4 = 3|0-4| +4 = 3×4 + 4 = 12+4 =16 > 14
Clearly, this checks out.
Try a number between 2/3 and 7 1/3. 1 fits this range.
3|x-4| + 4 = 3|1-4| + 4 = 3|-3| + 4 = 3×3 +4 = 9+4 = 13 < 14
And this is what we expect, since this is not part of our solution set.
Finally, try a number larger than 7 1/3. 8 fits this requirement.
3|x-4| + 4 = 3|8-4| + 4 = 3|4| + 4 = 12 + 4 = 16 > 14
which checks out.
This is NOT a complete check of our solution set, but it does give an indication that our answer is reasonable.
