(x-3)/(x+1) > 5
i solved and got x < -2
but this is not right once you plug it in. please help.
it should be x>-3
Not so. If x=-5/2, y=11/3 < 5
(x-3)/(x+1) > 5
if x > -1,
x-3 > 5(x+1)
x-3 > 5x+5
-8 > 4x
x < -2
but, we assumed x > -1, so that's a no-go
If x < -1,
x-3 < 5(x+1)
x-3 < 5x+5
-8 < 4x
x > -2
So, we have -2 < x < -1
opps sorry made some calculation mistakes....
To solve the inequality (x-3)/(x+1) > 5, we need to apply a specific method called "sign chart" or "interval notation." Let's go through the steps to solve this inequality correctly:
Step 1: Find the values of x for which the expression (x-3)/(x+1) equals zero.
To do this, set the numerator equal to zero and solve for x:
x - 3 = 0
x = 3
However, we should note that x = -1 would also make the denominator equal to zero, which would make the expression undefined. So we must exclude x = -1 from our solution set.
Step 2: Divide the number line into intervals using the critical points found in Step 1 (-1 and 3). These intervals are (-∞, -1), (-1, 3), and (3, ∞).
Step 3: Choose a test point from each interval to determine the sign of the expression (x-3)/(x+1).
For the interval (-∞, -1), we can choose x = -2 as our test point.
Substituting x = -2 into the expression:
((-2) - 3)/((-2) + 1) = (-5)/(-1) = 5
So, the expression is positive in this interval.
For the interval (-1, 3), we can choose x = 0 as our test point.
Substituting x = 0 into the expression:
((0) - 3)/((0) + 1) = (-3)/(1) = -3
So, the expression is negative in this interval.
For the interval (3, ∞), we can choose x = 4 as our test point.
Substituting x = 4 into the expression:
((4) - 3)/((4) + 1) = (1)/(5) = 1/5
So, the expression is positive in this interval.
Step 4: Analyze the sign chart to determine the solution.
From the sign chart, we can see that the expression is greater than 5 in the intervals (-∞, -1) and (3, ∞). However, in the interval (-1, 3), the expression is negative. Thus, the solution to the inequality is x < -1 or x > 3.
Therefore, x < -1 is the correct solution, rather than x < -2.