How to solve this (2x+5)/(x+1) > (x+1)/(x-1)
and graph?
I tried this and got (x+3)(x-2)
and got -3<x>2 which isn't right
the right answer is x<-3 or -1<x<1 or x>2
How would I get this?
I asked this before.
I was sure that I answered this question before, but I can't find it in the postings.
I did find the sheet of paper with the solution on it
Here is what I did ...
(2x+5)/(x+1) > (x+1)/(x-1)
(2x+5)/(x+1) - (x+1)/(x-1) > 0
[2x^2 + 3x - 5 - x^2 - 2x - 1]/[(x+1)(x-1)] > 0
(x^2 + x - 6)/(x^2 - 1) > 0
(x+3)(x-2)/[(x+1)(x-1] > 0
clearly x ≠ ±1
so we have "critical" values at x= -3,-1,+1, and 2
Mark these values on a number line, I then pick any number in each of these 5 regions bordered by the above points. You don't have to get the actual numercal value, just the sign of the answer.
1. let x = -5
(-)(-)/(-)(-) which is > 0 . Good one!
2. let x = -2
(+)(-)/(-)(-) < 0 , no good
3. let x = 0
(+)(-)/(+)(-) > ) , Good one
4. let x = 1.5
(+)(-)/(+)(+) < 0 , no good
5. let x = 10
(+)(+)/(+)(+) > 0 Good one!
so x < -3 OR -1 < x < 1 OR x > 2
but for (-)(-)/(-)(-)
wouldn't it become +
does the answer have to be a negative for it to be true ?
Or would it having to be negative only for polynomial inequalities and for rational inequalities they have t be positive?
It is positive or negative according to the stated inequality.
In your case after I moved the term to the left I was left with > 0, which means positive.
So we were looking for cases where the result is positive
That is why for case 1.
(-)(-)/(-)(-)
which is > 0 , or positive, I said "Good one"
To solve the inequality (2x+5)/(x+1) > (x+1)/(x-1), we need to follow a systematic approach. Let's break it down step by step:
Step 1: Find the restrictions
First, we need to identify any restrictions on the variable x. In this case, the expression (x+1)/(x-1) is undefined when the denominator, (x-1), equals zero. Therefore, x cannot be equal to 1.
Step 2: Determine the domain
Next, we determine the domain of the inequality. Since the denominators (x+1) and (x-1) cannot equal zero, x cannot be equal to -1 or 1.
Step 3: Determine the critical points
To find the critical points, we set each side of the inequality to zero and solve for x.
For the left side, we have (2x+5)/(x+1) > 0. Setting 2x+5 = 0, we get 2x = -5, and x = -5/2.
For the right side, we have (x+1)/(x-1) > 0. Setting x+1 = 0, we get x = -1. However, we already found that x = -1 is a restriction, so we exclude it from our critical points.
Step 4: Create a sign chart
Now, we create a sign chart to analyze the inequality in different intervals. Our critical points are -5/2 and 1, but since we know that x cannot be equal to -1 or 1, we divide our real number line into four intervals: x < -5/2, -5/2 < x < -1, -1 < x < 1, and x > 1.
Step 5: Analyze the intervals
Now, we need to determine the sign of the inequality in each of the intervals. We can do this by selecting test values within each interval and checking the sign of the inequality.
For x < -5/2, we could choose x = -3. Plugging this value into the inequality, we get (2(-3)+5)/(-3+1) = (-1)/(-2) = 1/2, which is positive. So, the inequality is true in this interval.
For -5/2 < x < -1, we could choose x = -2. Plugging this value into the inequality, we get (2(-2)+5)/(-2+1) = (1)/(-1), which is negative. So, the inequality is false in this interval.
For -1 < x < 1, we could choose x = 0. Plugging this value into the inequality, we get (2(0)+5)/(0+1) = (5)/(1) = 5, which is positive. So, the inequality is true in this interval.
For x > 1, we could choose x = 2. Plugging this value into the inequality, we get (2(2)+5)/(2+1) = (9)/(3) = 3, which is positive. So, the inequality is true in this interval.
Step 6: Write the solution
Based on our sign chart, we see that the inequality is true when x is less than -5/2, between -1 and 1, or greater than 1. Therefore, the solution is x < -5/2 or -1 < x < 1 or x > 1.
To graph the solution, we can shade the regions on a number line indicated by the intervals x < -5/2, -1 < x < 1, and x > 1. This shade will cover all the values of x that satisfy the inequality.