Solve. (x+5)(x-3)(x+4) is greater than 0.
Use at least one inequality or compound inequality to express your answer. Type R if the answer is in all real numbers.
The expression is zero at x= -5, -4 and +3. It becomes increasingly negative for x<-5 and increasing positive for x>3.
It is negative for -4<x<3 and positive for -5<x<-4
Therefore the answer is x>3 and -5<x<-4
Try graphing it and see. It is negative at x=0 and positive for x=-4.5,so that works.
To determine the solution to the inequality (x+5)(x-3)(x+4) > 0, we need to find the intervals where the expression is positive.
First, let's find the critical points by setting each factor equal to zero:
x + 5 = 0 => x = -5
x - 3 = 0 => x = 3
x + 4 = 0 => x = -4
These critical points divide the number line into four intervals:
Interval 1: x < -5
Interval 2: -5 < x < -4
Interval 3: -4 < x < 3
Interval 4: x > 3
To determine the sign of the expression in each interval, we can pick a test point from each interval and substitute it into the expression. Here, I'll use -6, -4.5, 0, and 4 as the test points for each interval, respectively.
For Interval 1: (-6+5)(-6-3)(-6+4) = (-1)(-9)(-2) = 18 > 0
For Interval 2: (-4.5+5)(-4.5-3)(-4.5+4) = (0.5)(-7.5)(-0.5) = 1.875 < 0
For Interval 3: (0+5)(0-3)(0+4) = (5)(-3)(4) = -60 < 0
For Interval 4: (4+5)(4-3)(4+4) = (9)(1)(8) = 72 > 0
From the signs of the expression in each interval, we can conclude that the solution to the inequality (x+5)(x-3)(x+4) > 0 is:
x < -5 or 3 < x
So, the answer is x < -5, x > 3, or in interval notation, (-∞, -5) ∪ (3, ∞).
To solve the inequality (x+5)(x-3)(x+4) > 0, we need to consider the sign of each factor and find the intervals where the product is positive.
1. Set each factor equal to zero and solve for x:
x + 5 = 0 --> x = -5
x - 3 = 0 --> x = 3
x + 4 = 0 --> x = -4
2. Plot these values on a number line:
-∞ -5 -4 3 +∞
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3. Test a value in each interval to determine the sign of the product:
- Substitute x = -6 (any value less than -5) into the inequality:
(-6 + 5)(-6 - 3)(-6 + 4) = (-1)(-9)(-2) = -18 < 0 (negative)
- Substitute x = 0 into the inequality:
(0 + 5)(0 - 3)(0 + 4) = (5)(-3)(4) = -60 < 0 (negative)
- Substitute x = 1 (any value between -4 and 3) into the inequality:
(1 + 5)(1 - 3)(1 + 4) = (6)(-2)(5) = -60 < 0 (negative)
- Substitute x = 10 (any value greater than 3) into the inequality:
(10 + 5)(10 - 3)(10 + 4) = (15)(7)(14) = 1470 > 0 (positive)
4. Based on the test values, we can see that the product is negative in the intervals (-∞, -5), (-4, 3), and (0, +∞). However, it is positive in the interval (-5, -4).
Therefore, the solution to the inequality (x+5)(x-3)(x+4) > 0 is -5 < x < -4.