How to solve this inequality and graph?
(2x+5)/(x+1) > (x+1)/(x-1)
would I first get rid of x-1 and divide x+1 over to get
(2x+5)(x-1)/(x+1)(x+1) > 0
if so, then who would I graph this?
To solve the inequality (2x+5)/(x+1) > (x+1)/(x-1), you correctly started by eliminating the denominators by multiplying both sides of the inequality by (x+1)(x-1). However, there is a small mistake in the expression you wrote.
Multiplying both sides of the inequality by (x+1)(x-1), we get:
(2x+5)(x-1) > (x+1)(x+1)
Now, let's simplify this inequality step by step:
2x^2 - 2x + 5x - 5 > x^2 + x + x + 1
2x^2 + 3x - 5 > x^2 + 2x + 1
Next, let's combine like terms on both sides:
2x^2 - x^2 + 3x - 2x > 1 + 5
x^2 + x > 6
Now, to graph this inequality, we need to find the values of x that make the inequality true. To do this, we will first find the critical points by setting the inequality equal to zero:
x^2 + x - 6 = 0
Factoring the quadratic equation, we have:
(x + 3)(x - 2) = 0
Thus, the critical values of x are x = -3 and x = 2.
Now, let's make a sign chart:
Interval (-∞, -3) (-3, 2) (2, ∞)
Test Value x = -4 x = 0 x = 3
x^2 + x > 6 + + -
From the sign chart, we can see that the inequality is satisfied for x < -3 and x > 2. Thus, the solution to the inequality is:
x < -3 or x > 2
To graph this inequality, you would plot an open circle at x = -3 and another open circle at x = 2 to represent that these values are not included in the solution. Then, shade the regions to the left of -3 and to the right of 2 on the number line to represent the solution set.