Solve the nonlinear inequality. Express the solution using interval notation.
(x − 6)(x − 5)(x + 4) > 0
To solve the inequality (x - 6)(x - 5)(x + 4) > 0, we need to find the intervals where the expression is positive.
First, let's determine the critical points by setting each factor equal to zero:
x - 6 = 0 => x = 6
x - 5 = 0 => x = 5
x + 4 = 0 => x = -4
These critical points divide the number line into four intervals: (-∞, -4), (-4, 5), (5, 6), and (6, +∞).
Now, we can test a number from each interval to see if the expression is positive or negative.
For the interval (-∞, -4), we can use x = -5:
(-5 - 6)(-5 - 5)(-5 + 4) > 0
(-11)(-10)(-1) > 0
The expression is negative in this interval.
For the interval (-4, 5), we can use x = 0:
(0 - 6)(0 - 5)(0 + 4) > 0
(-6)(-5)(4) > 0
The expression is positive in this interval.
For the interval (5, 6), we can use x = 5.5:
(5.5 - 6)(5.5 - 5)(5.5 + 4) > 0
(-0.5)(0.5)(9.5) > 0
The expression is negative in this interval.
For the interval (6, +∞), we can use x = 7:
(7 - 6)(7 - 5)(7 + 4) > 0
(1)(2)(11) > 0
The expression is positive in this interval.
From the above analysis, we can conclude that the solution to the inequality is:
(-4, 5)U(6, +∞)
To solve the inequality (x − 6)(x − 5)(x + 4) > 0, we can use the concept of intervals.
First, let's find the critical points by setting each factor equal to zero:
x - 6 = 0
x - 5 = 0
x + 4 = 0
Solving these equations, we find the critical points:
x = 6
x = 5
x = -4
Now, we will create a number line and plot these critical points on it:
-∞ -4 5 6 +∞
Next, we need to test intervals between the critical points to determine where the inequality is true. We can pick any number within each interval and substitute it into the inequality to see if it holds true.
Let's test the interval (-∞, -4):
Pick x = -5:
(-5 - 6)(-5 - 5)(-5 + 4) > 0
(-11)(-10)(-1) > 0
This is false, so the inequality is not true in the interval (-∞, -4).
Next, let's test the interval (-4, 5):
Pick x = 0:
(0 - 6)(0 - 5)(0 + 4) > 0
(-6)(-5)(4) > 0
This is true, so the inequality is true in the interval (-4, 5).
Finally, let's test the interval (6, +∞):
Pick x = 7:
(7 - 6)(7 - 5)(7 + 4) > 0
(1)(2)(11) > 0
This is true, so the inequality is true in the interval (6, +∞).
Therefore, the solution to the inequality (x − 6)(x − 5)(x + 4) > 0 is:
(-4, 5) U (6, +∞).