Download Connections Applications of Inequalities

Applications of
Inequalities
Connections
Have you ever . . .
• Saved money to pay for a vacation or a car?
• Read a manual to find the most weight a truck could hold?
• Figured out the most you could spend on clothes?
Often, a solution is not limited to a single value, but consists of
a range of values. You might need to spend less than a certain
amount on groceries, keep your load to less than the maximum
your truck can carry, or save at least enough for that trip to
Hawaii. You can use inequalities to solve these types of problems.
Unlike equations, inequalities have a range of possible solutions. The solution is a range of
values that makes the inequality true. The maximum weight allowed in an elevator is usually 3,400 pounds, but does that mean it is the only weight allowed? Of course not. Anything
below or equal to 3,400 pounds is allowed. The inequality sign to describe this limit is the
less than or equal to sign (#).
weight allowed # 3400 pounds
The weight allowed on the elevator is less than or equal to 3400 pounds. Instead of an equal
sign, an inequality uses one of the following signs:
>
<
is greater than
is less than
$ is greater than or equal to
# is less than or equal to
You can show the range of values of an inequality by graphing it on a number line.
Y
–5 –4 –3 –2 –1 0
Y
1
2
3
4
Y
5
–5 –4 –3 –2 –1 0
Y
1
2
3
4
5
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Essential Math Skills
Learn
It!
Solving Inequalities with Undo
Even though inequalities have more than one solution, you can still solve them
by using undo. There is one big difference. When multiplying or dividing both
sides by a negative number, flip the direction of the inequality sign.
Why? Examine this inequality:
-2 < 3
The statement is true, but if you multiply or divide by a negative number, you change the
signs and make the statement untrue. You must flip the inequality sign so that the statement
remains true.
-2 ÷ -1 < or > 3 ÷ -1 ? False  2 # -3 True  2 > -3
–5 –4 –3 –2 –1 0
1
2
3
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5
–5 –4 –3 –2 –1 0
1
2
3
4
5
A stock started out at $8.24 per share. The next two days, it went down by the
same amount each day. Elias wants to buy the stock if it’s less than or equal to
$5 per share. What drop in price for two days in a row would cause the stock to be
$5 per share or less? Graph the answer on a number line.
Un Understand
When you approach an inequality, determine what inequality sign you will need to use.
?
1. What inequality sign will you use in this problem?
Since Elias wants to buy the stock if it is less than or equal to $5, you should use the less than
or equal to inequality sign (#).
P Plan
When you plan, write an inequality. Writing an inequality is similar to writing an equation.
Pay special attention to the correct inequality sign.
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Applications of Inequalities
?
2. Write an inequality to represent the problem.
Let x = the drop in price each day. You might word this inequality, “Eight dollars and twentyfour cents minus twice the drop in price each day is less than or equal to five dollars.”
8.24 - 2x # 5
A Attack
Build Your
Math Skills
To undo an inequality:
• Undo parentheses by simplifying what’s in the parentheses
and distributing multiplication.
Distributing a
number multiplied
by parentheses
means you multiply
the number by
each term in the
parentheses.
• Multiply and divide constants with terms.
• Combine like terms.
• Use inverse operations to undo addition and subtraction.
• Use inverse operations to undo multiplication and division.
Determine whether to flip the inequality sign.
?
2(2 + x) = 4 + 2x
3. Solve the inequality and graph the answer on a number line.
–12 –11 –10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0
1
2
3
4
5
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8
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10 11 12
Undo Addition and Subtraction
Since there are no parentheses, terms to multiply or divide, or like terms on the same side of
the inequality symbol, start by undoing addition and subtraction.
8.24 - 2x # 5 -8.24 -8.24 -2x # -3.24
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Essential Math Skills
Undo Multiplication and Division
When you undo multiplication or division in an inequality, determine whether to flip the
direction of the sign. Because you need to divide by a negative (–2), you will change the values. The equation is no longer true unless you flip the inequality sign.
-2x ÷ -2 $ -3.24 ÷ -2
x $ 1.62
If the price dropped by $1.62 or more each day, then it’s the right price to buy.
Graph
To graph a line that represents # or $, make a solid dot at the endpoint on the number line.
If the symbol is < or >, then make an open dot to indicate that the endpoint is not a solution.
Draw a line to the right for > or $ and to the left for < or #.
Y
–12 –11 –10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0
1
2
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5
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8
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10 11 12
C Check
To check an inequality, first write it as an equation. Then use the substitution process to
check your answer.
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4. Explain why the solution is correct.
Change the inequality sign to an equal sign and use substitution to check the solution:
8.24 - 2x = 5
8.24 - 2(1.62)= 5
8.24 - 3.24 = 5
5 = 5
The result is true, so the answer is correct.
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