Question

Solve for x under the assumption that x > 0. Enter your answer in interval notation using grouping symbols.

x−24/x<10

Answer 1

\[x-\frac{ 24 }{ x }<10\]

Answer 2

Let's find the critical points of the inequality.
x2−24/x=10
x2−24=10x (Multiply both sides by x)
x2−24−10x=10x−10x (Subtract 10x from both sides)
x2−10x−24=0
(x+2)(x−12)=0 (Factor left side of equation)
x+2=0 or x−12=0 (Set factors equal to 0)
x=−2 or x=12

Answer 3

1) Note that x is undefined at 0
x≠0

2) Replace the inequality sign with an equal sign,
so that we can solve it like a normal equation
x−24x=10

3) Multiply both sides by the common denominator x
x2−24=10x

4) Move all terms to one side
x2−24−10x=0

5) Factor x2−24−10x
1. Ask: Which two numbers add up to -10 and multiply to -24?
2. Answer: -12 and 2
3. Rewrite the expression as (x−12)(x+2):
(x−12)(x+2)=0

6) Solve for x
1. Ask: When will (x−12)(x+2) equal zero?
2. Answer: When x−12=0 or x+2=0.
3. Solve each of the 2 equations above:
x=12,−2

7) From the values of x above, we have these 4 intervals to test

x<−2−2<x<00<x<12x>12


8) Pick a test point for each interval

For the interval x<−2:

Let's pick x=−3. Then, −3−24−3<10.
After simplifying, we get 5<10, which is true.
Keep this interval.

For the interval −2<x<0:

Let's pick x=−1. Then, −1−24−1<10.
After simplifying, we get 23<10, which is false.
Drop this interval.

For the interval 0<x<12:

Let's pick x=1. Then, 1−241<10.
After simplifying, we get −23<10, which is true.
Keep this interval.

For the interval x>12:

Let's pick x=13. Then, 13−2413<10.
After simplifying, we get 11.1538461539<10, which is false.
Drop this interval.

9) Therefore,
x<−20<x<12

Hope this helps! :D

Answer 4

Check possible critical points.
x=−2(Works in original equation)
x=12(Works in original equation)
Critical points:
x=−2 or x=12(Makes both sides equal)
x=0(Makes left denominator equal to 0)
Check intervals in between critical points. (Test values in the intervals to see if they work.)
x<−2(Works in original inequality)
−2<x<0(Doesn't work in original inequality)
0<x<12(Works in original inequality)
x>12(Doesn't work in original inequality)

Answer 5

The answer shall be: Answer:
x<−2 or 0<x<12

Answer 6

thanks #ShootingStar28

Answer 7

You are welcome :)

Answer 8

can you help with another one? I think I know how to do it now, just want you to check my answer. I can put it as a new quetions, if you want

Answer 9

\[x-\frac{ 15 }{ x }<-2\]

Answer 10

Add
\[\frac{ 15 }{ 2 }\] to both sides

\[x<-2+\frac{ 15 }{ 2 }\]

Then simplify

\[-2+\frac{ 15 }{ 2 }\] to \[\frac{ 11 }{ 2 }\]

So, \[x<\frac{ 11 }{ ? }\]

Hope this helps! :)

Answer 11

Oops, I meant \[x=\frac{ 11 }{ 2 }\]

Answer 12

Ugh, Sorry

\[x<\frac{ 11 }{ 2 }\]