Solve the following Inequality

Question

(x+4)^2(x-6)>0

What is the solution?

I forgot the steps to solve this and my notes are a bit sloppy. I would really appreciate if someone can walk me through this step by step so I won't have this issue in the future.

Thank you.

Answer 1

Here's one way to solve (x+4)²·(x-6) > 0:

Step 1 Solve (x+4)²·(x-6) = 0

Hopefully, solving (x+4)²·(x-6) = 0 will not be difficult. Since the left-hand side of this equation is already factored, we just need to set each factor equal to zero and solve for x.

(x + 4)² = 0 <---> x = -4

x - 6 = 0 <---> x = 6

Step 2 With these two values x = -4 and x = 6, we divide the number line into three sections

Section 1: all values x < -4

Section 2: all values -4 < x < 6

Section 3: all values x > 6

Step 3 We now choose one test value of x for each section. If that value satisfies the inequality (x+4)²·(x-6) > 0, then that section is part of our solution. Keep in mind, any value from that section will due.

Section 1: Let x = -5. Then (-5+4)²(-5-6) = (1) (-11) is negative. NOT A SOLUTION

Section 2: Let x = 0. Then (0+4)²(0-6) = (16)(-6) is negative. NOT A SOLUTION

Section 3: Let x = 7. Then (7+4)²(7-6) = (11²)(1) is positive. SOLUTION

Therefore, our solution lies in Section 3, that is all values of x greater than 6, {x | x > 6}

Answer 2

We want two terms, (x+4)² and (x-6), to multiply to something greater than zero. Two terms can multiply to a positive number in two ways: both terms are positive, or both terms are negative. Let's look at these two ways one at a time.

For both terms to be positive, we need (x+4)² > 0 and (x-6) > 0. But (x+4)² is always greater than or equal to zero, since it is a square. The one value of x for which this term is not greater than zero is when it is equal to zero, which happens when x=-4. So the first term is positive for all values of x not equal to -4. The second term is greater than zero for x>6. So for both of these terms to be positive, we need x>6.

For both terms to be negative, (x+4)² would need to be negative, which is impossible! So both terms can't be negative. So there are no values of x that make both terms negative.

So the solution to this inequality is x>6. When x>6, both terms are positive, so they multiply to something greater than zero. Hope this helps!