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Question 1194429: The numerator of the fraction is x, while the denominator is 5 more the numerator if the value of the fraction is less than 2/5, find the value of x
Found 2 solutions by math_tutor2020, ikleyn:
Answer by math_tutor2020(3816)   (Show Source):
You can put this solution on YOUR website!
x = numerator
x+5 = denominator
x/(x+5) is the fraction itself
It is stated to be less than 2/5
x/(x+5) < 2/5
Notice that x = -5 leads to a division by zero error.
This is a natural cutting point to divide the x axis number line into two pieces, and we'll test each piece.
Let's plug in something to the left of x = -5, let's say x = -6
So,
x/(x+5) < 2/5
-6/(-6+5) < 2/5
6 < 2/5
6 < 0.4
which is clearly false
You'll find a false statement results when you plug in anything to the left of x = -5
So we ignore x < -5 and x = -5 itself.
Let's try something to the right of x = -5. I'll try x = 0
x/(x+5) < 2/5
0/(0+5) < 2/5
0 < 2/5
0 < 0.4
That is true. This suggests that x > -5 is the solution set.
However, we need to solve for x like so
x/(x+5) < 2/5
5x < 2(x+5)
5x < 2x+10
5x-2x < 10
3x < 10
x < 10/3
The inequality sign stays the same the entire time. It only flips if we were to multiply or divide both sides by a negative number.
The reality is that yes x > -5 must be the case, but we also need x < 10/3 as well
Put together, the solution set is -5 < x < 10/3
A quick way to verify this is to graph these equations
y = x/(x+5)
y = 2/5
Then notice how the hyperbolic curve is below the horizontal line along the interval -5 < x < 10/3
Graph:
https://www.desmos.com/calculator/fmus5i4ao0
Desmos is a free graphing app.
You can click on the intersection of the horizontal line and the curve to see the coordinates of said intersection point.
Keep in mind that 10/3 = 3.333 approximately.
Answer: Anything on the interval -5 < x < 10/3
Answer by ikleyn(52778)  (Show Source):
You can put this solution on YOUR website! .
The numerator of the fraction is x, while the denominator is 5 more the numerator.
if the value of the fraction is less than 2/5, find the value of x
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Let x be the numerator; then the denominator is (x+5) and the fraction is  .
They want you solve this inequality
<  . (1)
+---------------------+
| Attention (!) |
+---------------------+
Do not multiply both sides by (x+5), because you don't know in advance,
if this value is positive or negative.
+----------------------------------------------------------------------+
| THERE IS ANOTHER METHOD TO SOLVE IT without making wrong steps ! |
+----------------------------------------------------------------------+
Re-write equation (1) equivalently in this form
- < 0.
Write it with the common denominator, which is 5*(x+5).
 -  < 0.
Simplify
 < 0,
 < 0,
 < 0.
This inequality is equivalent to
 < 0.
The last inequality is hold in these two cases:
(a) 3x-10 < 0 and x+5 > 0
or
(b) 3x-10 > 0 and x+5 < 0.
In case (a), it implies 3x < 10 and x > -5, which is the same as x < 10/3 and x > -5,
which gives the answer -5 < x < 10/3.
In case (b), it implies 3x > 10 and x < -5, which has no solutions.
ANSWER. The solution set is -5 < x < 10/3, or, in interval form, (-5, 10/3).
Solved.
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