Solve the nonlinear inequality. Express the solution using interval notation.
5x + 15
x − 6
< 0
To solve the inequality, we need to find the values of x for which the expression (5x + 15)/(x - 6) is less than 0.
First, let's find the critical points by setting the expression equal to zero and solving for x:
(5x + 15)/(x - 6) = 0
5x + 15 = 0
5x = -15
x = -3
So, x = -3 is a critical point.
Now, we can test intervals to determine the signs of the expression.
For x < -3: Choose x = -4
(5(-4) + 15)/(-4 - 6) = -5/(-10) = 1/2
Since 1/2 > 0, this interval does not satisfy the inequality.
For -3 < x < 6: Choose x = 0
(5(0) + 15)/(0 - 6) = 15/-6 = -5/2
Since -5/2 < 0, this interval satisfies the inequality.
For x > 6: Choose x = 7
(5(7) + 15)/(7 - 6) = 50/1 = 50
Since 50 > 0, this interval does not satisfy the inequality.
Therefore, the solution to the inequality is x ∈ (-3, 6).
To solve the inequality \(\frac{{5x + 15}}{{x - 6}} < 0\), we can follow these steps:
Step 1: Find the critical points by setting the numerator and denominator equal to zero:
\(x - 6 = 0\) (denominator)
\(x = 6\)
Step 2: Create a number line with the critical points and test points:
Choose a test point in each interval created by the critical point(s). Let's choose -1, 0, and 7.
Step 3: Determine the sign of the inequality in each interval:
For \(x < 6\):
Substitute -1 into the inequality:
\(\frac{{5(-1) + 15}}{{-1 - 6}} < 0\) simplifies to \(-\frac{{10}}{{7}} < 0\), which is true.
So, the inequality is true for \(x < 6\).
For \(x > 6\):
Substitute 7 into the inequality:
\(\frac{{5(7) + 15}}{{7 - 6}} < 0\) simplifies to \(\frac{{50}}{{7}} < 0\), which is false.
So, the inequality is false for \(x > 6\).
Step 4: Express the solution in interval notation:
The solution to the inequality is \(x < 6\).
In interval notation, this can be written as \((-∞, 6)\).