2.7LinearInequalities
There are two ways we can examine linear inequalities. We can solve an inequalities if there is only one
variable graphing the solution set on a number line, or if given two variables we can graph the solution set on the
coordinate plane.
One Variable Inequalities
Solving one variable inequalities is just like solving equations. We use inverse operations. However, there
is something odd that happens in particular cases. Let’s take a look at a couple of examples.
2 − 17 f −9
2 − 17 + 17 f −9 + 17
2 f 8
2 8
f
2
2
f4
According to our work, any number less than four should be a solution. Check this by plugging in a simple
value less than four, say zero, to see if this is actually true.
20 − 17 = −17 f −9
That statement is true. Therefore we can be confident that we have the correct solution. Now we graph
the solution set f 4 on a number line remembering that less than has an open circle and less than or equal to has
a closed circle.
f4
This is just a visual way to see that any number less than four is a solution to our inequality.
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Now let’s just slightly modify the original problem by making the coefficient negative and see what
happens.
−2 − 17 f −9
−2 − 17 + 17 f −9 + 17
−2 f 8
8
−2
f
−2
−2
f −4
According to our work here, any value less than negative four should be a correct solution. Let’s check
the value of −5 to see if it works.
−2−5 − 17 = −7 f −9
This statement is not true. Negative five did not work as a solution. What went wrong? It has to do with
the negative coefficient on . Let’s solve that last step in another way.
−2 f 8
−2 + 2 f 8 + 2
0 f 2 + 8
0 − 8 f 2 + 8 − 8
−8 f 2
−8 2
f
2
2
−4 f This looks like the same solution, but let’s turn it around how we normally see it. If negative four is less
than , then is greater than negative four, or > −4. This is the opposite sign that we got originally. Let’s check
to see if this works by picking a number greater than negative four, say zero, and plug it in to see if it makes a true
statement.
−20 − 17 = −17 f −9
This is a true statement. So our original work had the sign turned around. It looks like inverse operations
didn’t work. However, we make sure that we have the sign the right way by making sure that we change the
inequality sign anytime that we multiply or divide by a negative number.
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Two Variable Inequalities
Solutions to two variable inequalities is best represented on a coordinate plane. We can put the inequality
in slope-intercept form, graph the associated line, and then graph the solution set. Let’s work through an example
problem to see the notation and results. We’ll start by putting our inequality in slope-intercept form.
4 2 g 6
4 4 2 g 6 4
2 g 4 6
2 4 6
g
2
2
g 2 3
Now we’ll pretend like that is an equals sign instead of an inequality sign and graph that line as follows:
Since we’re dealing less than OR EQUAL TO, we’ll use a solid line. If it
were just less than, we would use a dotted line to show that any point on the
line would not be a solution.
Now think about what each variable means in our inequality. The is
represented vertically, so our inequality really says the height is less than or
equal to the line. So pick any point on the line and go below it. That will be the
area of the coordinate plane that we want to shade.
Graphically, this means that if you pick any point below that line and
substitute the and values into the equality, it will make a true statement.
The final graph looks like the one below.
Pick any point in the shaded area to double check. One easy point to
use is the origin, 0,0. Let’s plug in 0 and 0 in the original inequality.
40 20 0 g 6
This is a true statement, so we have the correct solution set shaded.
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Lesson 2.7
Solve the following inequalities and graph the solution on a number line.
1.
+3<7
3. 2 + 1 ≥ 11
2. − C + 4 ≤ 8
4. − − 2 > −2
5. − D f 2
6. 4 − 11 ≤ −3
7. 2 + 3 ≥ −13
8. − − 2 > −5
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Write and solve an inequality for each problem. Graph your solution on the number line.
9. Your middle school band is having a fundraiser selling boxes of Krispy Kreme donuts in order to purchase new
marching uniforms. They spent $3,500 advertising their fundraiser and make $1.75 per box sold. How many boxes
do they need to sell in order to at least break even?
10. The gymnasium can legally hold 1,250 people. During graduation, they set up rows of 15 chairs and have 50
chairs set aside for the faculty. How many rows could they put up in the gym?
11. Every study session at home should be around 20 minutes long. You know that you will spend class time
studying totaling 2 hours exactly. How many study sessions at home should you have if you want to spend at least
5 hours studying?
12. For every 10 box tops, the school library gets $1 to buy new books for you to read. The school spends $200 on
prizes for the box top competition. How many box tops need to be brought in if the library wants to purchase
$1,200 of new books?
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Graph the solution set to each two variable inequality on the coordinate plane provided.
13. < 2 1
14. g 4
15. k 2 1
16. / C 4
17. g 3 7
18. k 2
C
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19. 2 f 4
20. 3 2 g 4
21. k 1
22. C C / 1
23. 3 2 g 8
24.
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C
2 k 6