SOLUTION: Solve the following inequality. 40x-84<40/x a.) Is the point x=0 included in the solution set of the inequality? b.)Are the other finite end points of the intervals included i

Algebra ->  Rational-functions -> SOLUTION: Solve the following inequality. 40x-84<40/x a.) Is the point x=0 included in the solution set of the inequality? b.)Are the other finite end points of the intervals included i      Log On


   

Question 1065357: Solve the following inequality.
40x-84<40/x
a.) Is the point x=0 included in the solution set of the inequality?
b.)Are the other finite end points of the intervals included in the solution set?
c.) What is the solution set?
Found 2 solutions by josgarithmetic, ikleyn:
Answer by josgarithmetic(39625) About Me  (Show Source):
You can put this solution on YOUR website!
Posting better solution, more complete than earlier one -
-
x cannot be 0, and 0 is one of the critical x-values.

Divide both sides by 10
10x-21%3C10%2Fx
10x-21-10%2Fx%3C0

Concentrate on the expression on the left side
%28x%2Fx%2910x-%28x%2Fx%2921-%28x%2Fx%29%2810%2Fx%29

%2810x%5E2-21x%29%2Fx-10%2Fx

%2810x%5E2-21x-10%29%2Fx

Look for the critical x-values of the numerator; the denominator has already been taken care of.

Critical x-values are %2821-sqrt%2821%5E2%2B4%2A4%2A10%2A10%29%29%2F%282%2A10%29 and %2821%2Bsqrt%2821%5E2%2B4%2A4%2A10%2A10%29%29%2F%282%2A10%29
OR after computing,
system%28-2%2F5%2Cand%2C2%261%2F2%29.

The three critical values cut the x-axis into four intervals on which to check for solutions.
Check within each interval using the original inequality.
Pick any value within each interval.
-infinity%3Cx%3C-2%2F5                  -2%2F5%3Cx%3C0              0%3Cx%3C2%261%2F2                   2%261%2F2%3Cx%3Cinfinity
pick -1                        pick -1/5                   pick 2                     pick 4

4%28-1%29-84%3C40%2F%28-1%29            40%28-1%2F5%29-84%3C40%2F%28-1%2F5%29       40%2A2-84%3C40%2F2                40%2A4-84%3C40%2F4

-88%3C-40                     -88%3C-20                   -4%3C20              76%3C10

TRUE                              TRUE                         TRUE                         FALSE

SOLUTION: The union of -infinity%3Cx%3C0 and 0%3Cx%3C2%261%2F2
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THIS SOLUTION POSTED EARLIER, INCOMPLETE AND WRONG:
40x-84-40%2Fx%3C0
10x-21-10%2Fx%3C0
x%2810x-21-10%2Fx%29%3C0
10x%5E2-21x-10%3C0
Critical x values, %2821%2B-+sqrt%2821%5E2%2B4%2A10%2A10%29%29%2F20
%2821%2B-+sqrt%2821%5E2%2B400%29%29%2F20
%2821%2B-+sqrt%28841%29%29%2F20
%2821%2B-+29%29%2F20
These critical x values are -2/5 and 2.5.
You will find that the solution is the x values BETWEEN BUT NOT INCLUDING -2/5 and positive 2.5.
-2%2F5%3Cx%3C2%261%2F2.*
*note: x%3C%3E0 because of the original inequality.

Answer by ikleyn(52847) About Me  (Show Source):
You can put this solution on YOUR website!
.
Solve the following inequality.
40x-84 < 40/x
a.) Is the point x=0 included in the solution set of the inequality?
b.)Are the other finite end points of the intervals included in the solution set?
c.) What is the solution set?
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Plots  y = 40x - 84 (red)  and  y = 40/x (green)


The inequality is hold where the red line is below  (lover than)  the green line.

The solution set is   x < -2%2F5  and   0 < x < 2.5.
In the interval notation the solution set is  (-infinity,-0.4) U (0,2.5).

The solution by  "josgarithmetic"  is wrong.  Ignore his writing.