Virginia SOL Algebra 2 - 2020 Edition
1.09 Review: Compound inequalities
Lesson

Recall what we have already seen about compound inequalities by considering each of these four different statements.

 "$x>1$x>1 AND $x<9$x<9" "$x<1$x<1 OR $x>9$x>9" "$x>1$x>1 OR $x<9$x<9" "$x<1$x<1 AND $x>9$x>9"

These are examples of compound inequalities, statements involving several inequalities.

Let's go through the compound inequalities one by one.

"$x>1$x>1 AND $x<9$x<9"

In other words, for any value of $x$x to satisfy this statement, it must be both greater than $1$1 and less than $9$9. This means $x$x will be a real number in between $1$1 and $9$9.

"$x<1$x<1 OR $x>9$x>9"

Either one of these inequalities could be true by themselves. $x$x could be less than $1$1, and it doesn't matter that it's not greater than $9$9, or vice versa. The only values of $x$x that don't satisfy this statement are numbers between $1$1 and $9$9 inclusive.

"$x>1$x>1 OR $x<9$x<9"

This involves the same inequalities as the first example, except the word "AND" has been replaced with the word "OR". How does this change the statement? Well, now it's okay for just one inequality to be true at a time. $x$x can either be greater than $1$1 or it can be less than $9$9. In fact, every possible number satisfies this condition. $x$x can be absolutely any value.

"$x<1$x<1 AND $x>9$x>9"

Is there any number that is both less than $1$1 and greater than $9$9? This is impossible! No number satisfies this compound inequality.

Because the last two aren't very interesting cases, we will mostly encounter examples like the first two.

Plotting Compound Inequalities

Previously, we covered how to plot inequalities on a number line.

How do we plot compound inequalities on the number line?

Let's say we have an example like the first one we saw above, say, "$x\ge-5$x5 AND $x\le-2$x2".

This includes every number in between $-5$5 and $-2$2, inclusive. We can plot it like this.

As you can see, while ordinary inequalities are plotted as rays, this kind of compound inequality is plotted as an interval.

Now, say we wanted to plot "$x\le-5$x5 OR $x\ge-2$x2". As we found before, this means that $x$x can either be less than or equal to $-5$5 OR greater than or equal to $-2$2. So, we plot both inequalities on the number line, as two rays.

Compound Inequalities written without "AND"

There is another way of writing any compound inequality like "$x\ge-5$x5 AND $x\le-2$x2".

The values of $x$x that satisfy this compound inequality include all numbers between $-5$5 and $-2$2, inclusive.

So, another way to write "$x\ge-5$x5 AND $x\le-2$x2" is:

$-5\le x\le-2$5x2

$-5$5 is less than or equal to $x$x, and $x$x is less than or equal to $-2$2. This also makes it easier to see that $x$x is in between $-5$5 and $-2$2!

Practice questions

Question 1

Consider the statement $3<6x+7<9$3<6x+7<9:

1. Is $3<6x+7<9$3<6x+7<9 a compound inequality?

No

A

Yes

B

No

A

Yes

B
2. What are the two inequalities that make up $3<6x+7<9$3<6x+7<9?

$\editable{}<6x+7$<6x+7

$\editable{}<\editable{}$<

3. What word joins the two inequalities that were obtained from the compound inequality $3<6x+7<9$3<6x+7<9?

OR

A

AND

B

OR

A

AND

B

QUESTION 2

Consider the compound inequality $x-5<6$x5<6 or $x+5>6$x+5>6.

1. Solve $x-5<6$x5<6 for $x$x.

2. Solve $x+5>6$x+5>6 for $x$x.

3. What is the solution to the compound inequality?

All $x$x such that $x<11$x<11.

A

All $x$x such that $11<x<11. B All$x$x such that$x>1$x>1. C All real$x$x. D All$x$x such that$x<11$x<11. A All$x$x such that$11<x<11.

B

All $x$x such that $x>1$x>1.

C

All real $x$x.

D
4. Plot the solution set on the number line below.

question 3

Consider the compound inequality $2x-18<-0.4$2x18<0.4 or $2x-18>0.4$2x18>0.4.

1. Solve $2x-18<-0.4$2x18<0.4 for $x$x.

2. Solve $2x-18>0.4$2x18>0.4 for $x$x.

3. Write the solution to the compound inequality.

4. Plot the solution set on the number line below.