Question

What is the solution to the quadratic inequality?

Answer 1

Answer:

First, solve each inequality. I'll solve the first one first.

7

≥

2

x

−

5

12

≥

2

x

6

≥

x

Therefore, x could be any number less than or equal to 6. In interval notation, this looks like:

(

−

∞

,

6

]

The parenthesis means that the lower end is not a solution, but every number above it is. (In this case, the lower end is infinity, so a parenthesis must be used, since infinity is not a real number and so it cannot be a solution.) The bracket means that the upper end is a solution. In this case, it indicates that not only could  

x

be any number less than 6, but it could also be 6.

Let's try the second example:

3

x

−

2

4

>

4

3

x

−

2

>

16

3

x

>

18

x

>

6

Therefore, x could be any number greater than 6, but x couldn't be 6, since that would make the two sides of the inequality equal. In interval notation, this looks like:

(

6

,

∞

)

The parentheses mean that neither end of this range is included in the solution set. In this case, it indicates that neither 6 nor infinity are solutions, but every number in between 6 and infinity is a solution (that is, every real number greater than 6 is a solution).

Now, the problem used the word "OR", meaning that either of these equations could be true. That means that either  

x

is on the interval  

(

−

∞

,

6

]

or the interval  

(

6

,

∞

)

. In other words,  

x

is either less than or equal to 6, or it is greater than 6. When you combine these two statements, it becomes clear that  

x

could be any real number, since no matter what number  

x

is, it will fall in one of these intervals. The interval "all real numbers" is written like this:

(

−

∞

,

∞

)

Final Answer

Step-by-step explanation: