Solving inequalities

Question

x^2+6x<or= -8

Answer 1

Hi Merna.

 

Move the -8 over to the left side of the equation.

 

x² + 6x + 8 ≤ 0

 

This factors very easily.

 

(x+4)(x+2) ≤ 0

 

Now, pretend for a second it's an equality. What values will make your term equal to 0?

 

x+4 = 0

x+2 = 0

 

x = -4, -2

 

These will be the points on your number line to evaluate.

 

 

__________|______________|_____________

                 -4                       -2

 

Choose a test point for each individual range and insert it into your factored inequality.

 

(x+4)(x+2) ≤ 0

 

What students tend to waste time on in these questions is finding out what the exact value of the test point is. But all you care about it whether it's positive or negative. When picking a test point to the left of -4, you can choose something as astronomical as -100000000 and it'll still be valid because -100000000 is to the left of -4.

 

(-100000000 + 4)(-100000000 +2)

 

The numbers in both parentheses are both obviously very negative. We don't care about the exact value, just that they're both negative. A negative times a negative is a positive, so to the left of 4 you'll have positive values.

 

++++++++|______________|_____________
                 -4                       -2

 

Now pick something between -4 and -2. -3 seems like the obvious choice.

 

(-3+4)(-3+2)

 

This is a positive times a negative. Which is a negative. Everything in between -4 and 2 will then be negative.

 

++++++++|-------------------|_____________
                 -4                       -2

 

Now pick something greater than -2. You can do something like 10000000000.

 

(10000000000+4)(10000000000+2)

 

These are both very positive numbers. Positive times positive is positive, so you'll have all positives to the right of -2.

 

++++++++|-------------------|+++++++++++
                 -4                       -2

 

Your original inequality was:

 

(x+4)(x+2) ≤ 0

 

So you're looking for x values which make your expression LESS THAN OR EQUAL TO 0. You want values on your number line which spit out negative results. These are values between -4 and -2. Everywhere else outputs positive results, which makes your inequality false.

 

Finally, since it's less than OR EQUAL TO, -4 and -2 are also valid.

 

[-4,-2]

 

Hope this helps.