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: 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 If c > 0, then ac 0, then ac < bc. If c bc. If c bc. Division Property If c > 0, then a  c b  c. Similar statements can be written for a b, and a > b. 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. Greater than symbol makes the arrow point to the right on the # line. Greater than symbol makes the arrow point to the right on the # line. Less than symbol makes the arrow point to the left on the # line. Less than symbol makes the arrow point to the left on the # line. If > or or or or <, then leave the circle open.

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

Ex 2. 4 – 3p > 16 – p Ex 2. 4 – 3p > 16 – p

Ex 3. 2y + 9 < 5y + 15 Ex 3. 2y + 9 < 5y y + 9 < 15 -3y < 6 Y > -2 I needed to flip the inequality because I divided by a negative number.

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) > /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 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. 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 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 and 3b + 7 < 1. 2b > 4 b > 2 and 3b < -6 b < -2 Graph both of the above inequalities. No Solution: There is no intersection.

III. Compound Inequalities Graph the solution of each compound inequality on a number line. Ex 3. 5x + 1 > 21 or 3x 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 4 or x – 2 < 2. x > -3or x < 4 Graph both of the above inequalities. ALL REALS !

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