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.
The lower limit is -7. We also need to mark it with a filled-in circle.
Where are they found on the graph? Yes! It is less than or equal to 9? What about –15? There are other numbers that satisfy both conditions.
It is also greater than or equal to -7? What about –15? No! Where are they found on the graph?
Because the word and is used, a number on the graph needs to satisfy both parts of the inequality
Yes! So let’s try 20. Does 20 satisfy both conditions?
No! So let’s try 20. Does 20 satisfy both conditions?
Since 20 does not satisfy both conditions, it can’t belong to the solution set
There is one region we have not checked
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.
Does 0 satisfy both conditions? Yes!
Does 0 satisfy both conditions? Yes!
If one number in a region completely satisfies an inequality, you can know that every number in that region satisfies the inequality.
Let’s graph another inequality:
tells us we want an open circle, The first sign First we mark the boundary points:
and the 12 tells us where the circle goes
and the 12 tells us where the circle goes
tells us we want a closed circle, The second sign
and the -1 tells us where the circle goes
The boundary points divide the line into three regions:
We need to test one point from each region No!Yes!
Notice that the word used is or, No!Yes! instead of and.
Or means that a number No!Yes! only needs to meet one condition.
Because –10 meets one condition, Yes! the region to which it belongs belongs to the graph.
Let’s check the next region: No!
Because –1 meets neither condition, No! the numbers in that region will not satisfy the inequality.
Now the final region: Yes!No!
Again, 15 meets one condition Yes! so we need to shade that region.
A quick review: Find and mark the boundary points. 2. Test points from each region. 3. Shade the regions that satisfy the inequality. ??? To graph a compound inequality:
A quick review: Find and mark the boundary points. 2. Test points from each region. 3. Shade the regions that satisfy the inequality. or
Given the graph below, write the inequality. First, write the boundary points
Then look at the marks on the graph, and write the correct symbol.
Since x is between the boundary points on the graph, it will be between the boundary points in the inequality.
Since x is between the boundary points on the graph, it will be between the boundary points in the inequality.
Try this one: Again, begin by writing the boundary points:
And again, you need to choose the correct symbols:
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.
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.
We will use the word, or, instead: Remember that or means a number has to satisfy only one of the conditions.
We will use the word, or, instead: Remember that or means a number has to satisfy only one of the conditions.
Is there any one number that belongs to both shaded sections in the graph? Say NO! NO!
So it would be incorrect to use and. And implies that a number meets both conditions.
... you remember that a compound inequality is just two inequalities put together. Solving compound inequalities is easy if...
You can solve them both at the same time:
Write the inequality from the graph: :Write boundaries:2:Write signs:3:Write variable:
Solve the inequality:Is this what you did?
You did remember to reverse the signs didn’t you?Good job!