Compound Inequalities

You already know inequalities. Often they are used to place limits on variables. That just means x can be any number equal to 9 or less than 9.

Sometimes we put more than one limit on the variable: Now x is still less than or equal to 9, but it must also be greater than or equal to –7.

Let’s look at the graph: The upper limit is 9. Because x can be equal to 9, we mark it with a filled-in circle. 5 10 15 -20 -15 -10 -5 -25 20 25

The lower limit is -7. We also need to mark it with a filled-in circle. 5 10 15 -20 -15 -10 -5 -25 20 25

There are other numbers that satisfy both conditions. Where are they found on the graph? What about –15? It is less than or equal to 9? Yes! 5 10 15 -20 -15 -10 -5 -25 20 25

Where are they found on the graph? What about –15? It is also greater than or equal to -7? No! 5 10 15 -20 -15 -10 -5 -25 20 25

Because the word and is used, a number on the graph needs to satisfy both parts of the inequality. 5 10 15 -20 -15 -10 -5 -25 20 25

So let’s try 20. Does 20 satisfy both conditions? Yes! 5 10 15 -20 -15 -10 -5 -25 20 25

So let’s try 20. Does 20 satisfy both conditions? No! 5 10 15 -20 -15 -10 -5 -25 20 25

Since 20 does not satisfy both conditions, it can’t belong to the solution set. 5 10 15 -20 -15 -10 -5 -25 20 25

There is one region we have not checked. 5 10 15 -20 -15 -10 -5 -25 20 25

We need to choose a number from that region. You want to choose 0? Good choice! 0 is usually the easiest number to work with. 5 10 15 -20 -15 -10 -5 -25 20 25

Does 0 satisfy both conditions? Yes! 5 10 15 -20 -15 -10 -5 -25 20 25

Does 0 satisfy both conditions? Yes! 5 10 15 -20 -15 -10 -5 -25 20 25

If one number in a region completely satisfies an inequality, you can know that every number in that region satisfies the inequality. 5 10 15 -20 -15 -10 -5 -25 20 25

Let’s graph another inequality: 5 10 15 -20 -15 -10 -5 -25 20 25

First we mark the boundary points: The first sign tells us we want an open circle, 5 10 15 -20 -15 -10 -5 -25 20 25

and the 12 tells us where the circle goes. 5 10 15 -20 -15 -10 -5 -25 20 25

and the 12 tells us where the circle goes. 5 10 15 -20 -15 -10 -5 -25 20 25

tells us we want a closed circle, The second sign tells us we want a closed circle, 5 10 15 -20 -15 -10 -5 -25 20 25

and the -1 tells us where the circle goes. 5 10 15 -20 -15 -10 -5 -25 20 25

The boundary points divide the line into three regions: 1 2 3 5 10 15 -20 -15 -10 -5 -25 20 25

We need to test one point from each region. No! Yes! 5 10 15 -20 -15 -10 -5 -25 20 25

Notice that the word used is or, instead of and. No! Yes! 5 10 15 -20 -15 -10 -5 -25 20 25

only needs to meet one condition. Or means that a number only needs to meet one condition. No! Yes! 5 10 15 -20 -15 -10 -5 -25 20 25

Because –10 meets one condition, the region to which it belongs . . . . . . belongs to the graph. Yes! 5 10 15 -20 -15 -10 -5 -25 20 25

Let’s check the next region: No! No! 5 10 15 -20 -15 -10 -5 -25 20 25

Because –1 meets neither condition, the numbers in that region will not satisfy the inequality. No! 5 10 15 -20 -15 -10 -5 -25 20 25

Now the final region: Yes! No! 5 10 15 -20 -15 -10 -5 -25 20 25

Again, 15 meets one condition so we need to shade that region. Yes! 5 10 15 -20 -15 -10 -5 -25 20 25

To graph a compound inequality: A quick review: To graph a compound inequality: 1. Find and mark the boundary points. 2. Test points from each region. 3. Shade the regions that satisfy the inequality. ? ? ? 5 10 15 -20 -15 -10 -5 -25 20 25

1. Find and mark the boundary points. A quick review: 1. Find and mark the boundary points. 2. Test points from each region. 3. Shade the regions that satisfy the inequality. or 5 10 15 -20 -15 -10 -5 -25 20 25

Given the graph below, write the inequality. First, write the boundary points. 5 10 15 -20 -15 -10 -5 -25 20 25

Then look at the marks on the graph, and write the correct symbol. 5 10 15 -20 -15 -10 -5 -25 20 25

Since x is between the boundary points on the graph, it will be between the boundary points in the inequality. 5 10 15 -20 -15 -10 -5 -25 20 25

Since x is between the boundary points on the graph, it will be between the boundary points in the inequality. 5 10 15 -20 -15 -10 -5 -25 20 25

Again, begin by writing the boundary points: Try this one: Again, begin by writing the boundary points: 5 10 15 -20 -15 -10 -5 -25 20 25

And again, you need to choose the correct symbols: 5 10 15 -20 -15 -10 -5 -25 20 25

Because the x-values are not between the boundary points on the graph, we won’t write x between the boundary points in the equation. 5 10 15 -20 -15 -10 -5 -25 20 25

Because the x-values are not between the boundary points on the graph, we won’t write them between the boundary points in the equation. 5 10 15 -20 -15 -10 -5 -25 20 25

We will use the word, or, instead: Remember that or means a number has to satisfy only one of the conditions. 5 10 15 -20 -15 -10 -5 -25 20 25

We will use the word, or, instead: Remember that or means a number has to satisfy only one of the conditions. 5 10 15 -20 -15 -10 -5 -25 20 25

Is there any one number that belongs to both shaded sections in the graph? NO! Say NO! 5 10 15 -20 -15 -10 -5 -25 20 25

So it would be incorrect to use and So it would be incorrect to use and. And implies that a number meets both conditions. 5 10 15 -20 -15 -10 -5 -25 20 25

Solving compound inequalities is easy if . . . . . . you remember that a compound inequality is just two inequalities put together.

You can solve them both at the same time:

Write the inequality from the graph: 5 10 15 -20 -15 -10 -5 -25 20 25 3: Write variable: 1: Write boundaries: 2: Write signs:

Is this what you did? Solve the inequality:

You did remember to reverse the signs . . . Good job! . . . didn’t you?