1.7 Introduction to Solving Inequalities Objectives: Write, solve, and graph linear inequalities in one variable. Solve and graph compound linear inequalities in one variable. Standards: 2.8.11.D Formulate inequalities to model routine and non-routine problems.

An inequality is a mathematical statement involving <, >, >, <, or .

Properties of Inequalities For all real numbers a, b, and c, where a < b: Addition Property a + c < b + c. Subtraction Property a – c < b – c. Multiplication Property Beginning with a ≤ c If c is positive, then ac < bc. If c is negative, then ac > bc. Division Property Beginning with a ≤ c If c is positive, then a  c < b  c. If c is negative, then a  c > b  c. Any value of a variable that makes an inequality true is a solution of the inequality.

II. Solve each inequality and graph the solution on the number line. If the inequality symbol opens towards the variable then shade to the right on the # line. (Example: x>2 or 5<x) If the inequality symbol opens away from the variable then shade to the left on the # line. (Example: x<4 or -3>x) If > or < , then shade in the circle. If >, < or  then leave the circle open.

II. Solve each inequality and graph the solution on the number line. Ex 1. 4x – 5 > 13

Ex 2. 4 – 3p > 16 – p

Ex 3. 2y + 9 < 5y + 15

Ex. 4 Claire’s test average in her world history class is 90 Ex. 4 Claire’s test average in her world history class is 90. The test average is 2/3 of the final grade and the homework is 1/3 of the final grade. What homework average does Claire need in order to have a final grade of at least a 93%? Final grade = 2/3 (test average) + 1/3 (homework average) 2/3 (90) + 1/3 (H) > 93 60 + 1/3 (H) > 93 1/3 (H) > 33 H > 99

III. Compound Inequalities – is a pair of inequalities joined by and or or. To solve an inequality involving AND, find the values of the variable that satisfy both inequalities. An AND compound inequality either has an answer because the inequalities INTERSECT or a no solution answer, because the inequalities DON’T INTERSECT. To solve an inequality involving OR, find those values of the variable that satisfy at least one of inequalities. An OR compound inequality either has an inequality solution because the inequalities DON’T INTERSECT or all real numbers because the inequalities INTERSECT and COVER THE ENTIRE NUMBER LINE.

III. Compound Inequalities Graph the solution of each compound inequality on a number line. Ex 1. 2x + 1 > 3 and 3x – 4 < 17

III. Compound Inequalities Graph the solution of each compound inequality on a number line. Ex 2. 2b – 3 > 1 and 3b + 7 < 1.

III. Compound Inequalities Graph the solution of each compound inequality on a number line. Ex 3. 5x + 1 > 21 or 3x + 2 < -1

III. Compound Inequalities Graph the solution of each compound inequality on a number line. Ex 4. x + 7 > 4 or x – 2 < 2.

Writing Activities: Solving Inequalities 11). Which Properties of Inequality differ from the corresponding Properties of Equality? Explain and include examples. 12). Why do the graphs of some inequalities include open circles, while others do not? Explain. 13). Describe two kinds of compound inequalities.

Homework Integrated Algebra II- Section 1.7 Level A Academic Algebra II- Section 1.7 Level B