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Creating Linear Inequalities. Introduction Inequalities 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.

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slide2

Introduction

Inequalities 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

key concepts continued
Key Concepts, continued

1.2.2: Creating Linear Inequalities in One Variable

slide5

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

slide6

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

key concepts continued1
Key Concepts, continued

1.2.2: Creating Linear Inequalities in One Variable

slide8

Guided Practice

Example 1

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?

1.2.2: Creating Linear Inequalities in One Variable

slide9

Guided Practice: Example 1, continued

Read the problem statement first.

Reread the scenario and identify the known quantities.

Juan has no more than $50.

Sales tax is 4%.

Juice costs $2.

Read the statement again, identify the unknown quantity.

cost of the jeans

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?

1.2.2: Creating Linear Inequalities in One Variable

slide10

Guided Practice: Example 1, continued

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

x+ 0.04x + 2 ≤ 50

1.2.2: Creating Linear Inequalities in One Variable

slide11

Guided Practice: Example 1, continued

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

    • 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

slide12

Guided Practice: Example 1, continued

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.

1.2.2: Creating Linear Inequalities in One Variable

guided practice example 1 continued
Guided Practice: Example 1, continued

1.2.2: Creating Linear Inequalities in One Variable

slide14

Guided Practice

Example 2

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?

1.2.2: Creating Linear Inequalities in One Variable

slide15

Guided Practice: Example 1, continued

Read the problem statement first.

Reread the scenario and identify the known quantities.

Needs at least $1,100

Saved $400

Makes $12 an hour

Read the statement again, identify the unknown quantity.

Least number of hours Alexis must work to make enough money

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?

1.2.2: Creating Linear Inequalities in One Variable

slide16

Guided Practice: Example 2, continued

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

400 + 12h ≥ 1100

1.2.2: Creating Linear Inequalities in One Variable

slide17

Guided Practice: Example 2, continued

Solve the problem.

400 + 12h ≥ 1100

Subtract 400 from both sides.

12h ≥ 700

Divide both sides by 12.

1.2.2: Creating Linear Inequalities in One Variable

slide18

Guided Practice: Example 2, continued

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.

1.2.2: Creating Linear Inequalities in One Variable

guided practice example 2 continued
Guided Practice: Example 2, continued

1.2.2: Creating Linear Inequalities in One Variable