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. 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 Symbol Description Example Solution > greater than, more than x > 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 < 3 all numbers less than 3; does not include 3 ≤ less than or equal to, no more than x ≤ 3 all numbers less than or equal to 3; includes 3 ≠ not equal to x ≠ 3 includes all numbers except 3 1.2.2: Creating Linear Inequalities in One Variable

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

Create an expression to represent each of the following: 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, continued Creating Inequalities from Context Read the problem statement first. Reread the scenario and make a list or a table of the known quantities. Read the statement again, identifying the unknown quantity or variable. Create expressions and inequalities from the known quantities and variable(s). Solve the problem. Interpret the solution of the inequality in terms of the context of the problem. 1.2.2: Creating Linear Inequalities in One Variable

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

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

Guided Practice: Example 1, continued Create expressions and inequalities from the known quantities and variable(s). x + 0.04x + 2 ≤ 50 Known Unknown 4% sales tax Cost of jeans Juice costs $2 No more than $50 1.2.2: Creating Linear Inequalities in One Variable

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 Add like terms. 1.04x + 2 ≤ 50 Subtract 2 from both sides. 1.04x ≤ 48 x ≤ 46.153846 Divide both sides by 1.04. 1.2.2: Creating Linear Inequalities in One Variable

✔ 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 http://walch.com/ei/CAU1L2S2ContextIneqSpend 1.2.2: Creating Linear Inequalities in One Variable

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

Least number of hours Alexis must work to make enough money 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

Guided Practice: Example 2, continued Create expressions and inequalities from the known quantities and variable(s). 400 + 12h ≥ 1100 Known Unknown Needs at least $1,100 Least 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 Solve the problem. Subtract 400 from both sides. 12h ≥ 700 Divide both sides by 12. 1.2.2: Creating Linear Inequalities in One Variable

✔ 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 http://walch.com/ei/CAU1L2S2ContextIneqSave 1.2.2: Creating Linear Inequalities in One Variable