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This lesson covers how to write and graph algebraic inequalities using examples. It also includes compound inequalities and their graphing.
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California Standards Preview of Grade 7 AF1.1 Use variables and appropriate operations to write an expression, an equation, an inequality, or a system of equations, or inequalities that represents a verbal description (e.g., three less than a number, half as large as area A).
Vocabulary inequality algebraic inequality solution set compound inequality
An inequalityis a statement that compares two expressions by using one of the following symbols: <, >, ≤, ≥, or . is less than Fewer than, below is greater than More than, above is less than or equal to At most, no more than is greater than or equal to At least, no less than
Additional Example 1: Writing Inequalities Write an inequality for each situation. A. There are at least 15 people in the waiting room. “At least” means greater than or equal to. number of people ≥ 15 B. The tram attendant will allow no more than 60 people on the tram. “No more than” means less than or equal to. number of people ≤ 60
Check It Out! Example 1 Write an inequality for each situation. A. There are at most 10 gallons of gas in the tank. “At most” means less than or equal to. gallons of gas ≤ 10 B. There is at least 10 yards of fabric left. “At least” means greater than or equal to. yards of fabric ≥ 10
An inequality that contains a variable is an algebraic inequality. A value of the variable that makes the inequality true is a solution of the inequality. An inequality may have more than one solution. Together, all of the solutions are called the solution set. You can graph the solutions of an inequality on a number line. If the variable is “greater than” or “less than” a number, then that number is indicated with an open circle.
This open circle shows that 5 is not a solution. a > 5 If the variable is “greater than or equal to” or “less than or equal to” a number, that number is indicated with a closed circle. This closed circle shows that 3 is a solution. b ≤ 3
Additional Example 2: Graphing Simple Inequalities Graph each inequality. Draw an open circle at 3. The solutions are values of n less than 3, so shade to the left of 3. A. n < 3 –3 –2 –1 0 1 2 3 B. a ≥ –4 Draw a closed circle at –4. The solutions are –4 and values of a greater than –4, so shade to the right of –4. –6 –4 –2 0 2 4 6
Check It Out! Example 2 Graph each inequality. Draw a closed circle at 2. The solutions are 2 and values of p less than 2, so shade to the left of 2. A. p ≤ 2 –3 –2 –1 0 1 2 3 B. e > –2 Draw an open circle at –2. The solutions are values of e greater than –2, so shade to the right of –2. –3 –2 –1 0 1 2 3
Writing Math The compound inequality –2 < y and y < 4 can be written as –2 < y < 4. A compound inequalityis the result of combining two inequalities. The words and and or are used to describe how the two parts are related. x > 3 or x < –1 –2 < y and y < 4 x is either greater than 3 or less than–1. y is both greater than –2 and less than 4. y is between –2 and 4.
–6 6 4 –4 –2 2 – – 2 4 6 0 0 – 6 4 2 – – – – – – 1 1 6 5 4 2 0 2 4 5 6 3 3 Additional Example 3: Graphing Compound Inequalities Graph each compound inequality. A. m ≤ –2 or m > 1 Graphm ≤ –2 Graph m > 1 • º Include the solutions shown by either graph.
–6 6 4 –4 –2 2 0 6 6 – – – 4 4 2 2 0 – – – – – – 1 1 6 5 4 2 0 2 4 5 6 3 3 Additional Example 3: Graphing Compound Inequalities Graph each compound inequality. B. –3 < b ≤ 0 Graph –3 < b Graph b ≤ 0 • º Include the solutions that the graphs have in common.
–6 6 4 –4 –2 2 – – 2 4 6 0 0 – 6 4 2 – – – – – – 1 1 6 5 4 2 0 2 4 5 6 3 3 Check It Out! Example 3 Graph each compound inequality. A. w < 2 or w ≥ 4 Graph w < 2 Graph w ≥ 4 Include the solutions shown by either graph.
–6 6 4 –4 –2 2 0 6 6 – – – 4 4 2 2 0 – – – – – – 1 1 6 5 4 2 0 2 4 5 6 3 3 Check It Out! Example 3 Graph each compound inequality. B. 5 > g ≥ –3 Graph 5 > g Graph g ≥ –3 • º Include the solution that the graphs have in common.
