Creating Linear Inequalities

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IntroductionInequalities are similar to equations in that they are mathematical sentences. They are different in that they are not equal all the time. An inequality has infinite solutions, instead of only having one solution like a linear equation.

Setting up the inequalities will follow the same process as setting up the equations did. Solving them will be similar, with two exceptions, which will be described later.The prefix in- in the word inequality means “not.” Remember that the symbols >, <, ≥, ≤, and ≠ are used with inequalities.

1.2.2: Creating Linear Inequalities in One Variable

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Key Concepts, continued

1.2.2: Creating Linear Inequalities in One Variable

Symbol Description Example Solution

>greater than,

more thanx > 3

all numbers greater than 3; doesn’t include 3

≥greater than or

equal to, at least x ≥ 3

all numbers greater than or equal to 3; includes 3

< less than x < 3all numbers less than 3;

does not include 3

≤less than or equal to, no more than

x ≤ 3all numbers less than or

equal to 3; includes 3

≠ not equal to x ≠ 3 includes all numbers except 3

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Vocabulary < >

Less than Greater than

At most At least

Fewer than

More than No more than

No less than

Under Over Maximum Minimum

Below Above

Key Concepts, continued• Solving a linear inequality is similar to solving a linear

equation.

• The processes used to solve inequalities are the same processes that are used to solve equations.

• Multiplying or dividing both sides of an inequality by a negative number requires reversing the inequality symbol.

1.2.2: Creating Linear Inequalities in One Variable

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Examples:Create an expression to represent each of the following: A number and 20 can be no more than 41

Four times some number is at most 16

The minimum value of a number is 84

45 is more than a number

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Key Concepts, continued

1.2.2: Creating Linear Inequalities in One Variable

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Creating Inequalities from Context

1. Read the problem statement first.2. Reread the scenario and make a list or a table of

the known quantities.3. Read the statement again, identifying the unknown

quantity or variable.4. Create expressions and inequalities from the

known quantities and variable(s).5. Solve the problem.6. Interpret the solution of the inequality in terms of

the context of the problem.

Guided Practice

Example 1Juan has no more than $50 to spend at the mall. He wants to buy a pair of jeans and some juice. If the sales tax on the jeans is 4% and the juice with tax costs $2, what is the maximum price of jeans Juan can afford?

1.2.2: Creating Linear Inequalities in One Variable

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Guided Practice: Example 1, continued

1. Read the problem statement first.

2. Reread the scenario and identify the known quantities.• Juan has no more than $50.• Sales tax is 4%.• Juice costs $2.

3. Read the statement again, identify the unknown quantity.

cost of the jeans

1.2.2: Creating Linear Inequalities in One Variable

Juan has no more than $50 to spend at the mall. He wants to buy a pair of jeans and some juice. If the sales tax on the jeans is 4% and the juice with tax costs $2, what is the maximum price of jeans Juan can afford?

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1.2.2: Creating Linear Inequalities in One Variable

Guided Practice: Example 1, continued

4. Create expressions and inequalities from the known quantities and variable(s).

x + 0.04x + 2 ≤ 5010

Known Unknown

4% sales tax

Cost of jeansJuice costs $2

No more than $50

1.2.2: Creating Linear Inequalities in One Variable

Guided Practice: Example 1, continued

5. Solve the problem.

Normally, the answer would be rounded down to 46.15. However, when dealing with money, round up to the nearest whole cent as a retailer would.

x ≤ 46.16

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• x + 0.04x + 2 ≤ 50• 1.04x + 2 ≤ 50• 1.04x ≤ 48• x ≤ 46.153846

Add like terms. Subtract 2 from both

sides. Divide both sides by

1.04.

1.2.2: Creating Linear Inequalities in One Variable

Guided Practice: Example 1, continued

6. Interpret the solution of the inequality in terms of the context of the problem.Juan should look for jeans that are priced at or below $46.16.

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1.2.2: Creating Linear Inequalities in One Variable

Guided Practice: Example 1, continued

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Guided Practice

Example 2Alexis is saving to buy a laptop that costs $1,100. So far she has saved $400. She makes $12 an hour babysitting. What’s the least number of hours she needs to work in order to reach her goal?

1.2.2: Creating Linear Inequalities in One Variable

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Guided Practice: Example 1, continued

1. Read the problem statement first.

2. Reread the scenario and identify the known quantities.• Needs at least $1,100• Saved $400• Makes $12 an hour

3. Read the statement again, identify the unknown quantity.Least number of hours Alexis must work to make enough money

1.2.2: Creating Linear Inequalities in One Variable

Alexis is saving to buy a laptop that costs $1,100. So far she has saved $400. She makes $12 an hour babysitting. What’s the least number of hours she needs to work in order to reach her goal?

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1.2.2: Creating Linear Inequalities in One Variable

Guided Practice: Example 2, continued

4. Create expressions and inequalities from the known quantities and variable(s).

400 + 12h ≥ 110016

Known Unknown

Needs at least $1,100Least number of hours Alexis must work to make enough money

Saved $400

Makes $12 an hour

1.2.2: Creating Linear Inequalities in One Variable

Guided Practice: Example 2, continued

5. Solve the problem.• 400 + 12h ≥ 1100

Subtract 400 from both sides.

• 12h ≥ 700 Divide both sides by 12.

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1.2.2: Creating Linear Inequalities in One Variable

Guided Practice: Example 2, continued

6. Interpret the solution of the inequality in terms of the context of the problem.In this situation, it makes sense to round up to the nearest half hour since babysitters usually get paid by the hour or half hour. Therefore, Alexis needs to work at least 58.5 hours to make enough money to save for her laptop.

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1.2.2: Creating Linear Inequalities in One Variable

Guided Practice: Example 2, continued

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