1.7 introduction to solving inequalities objectives: write, solve, and graph linear inequalities in...

1.7 Introduction to 1.7 Introduction to Solving InequalitiesSolving Inequalities

Objectives: Objectives: Write, solve, and graph Write, solve, and graph linear inequalities in one variable. linear inequalities in one variable. Solve and graph compound linear Solve and graph compound linear

inequalities in one variable.inequalities in one variable.

Standards: Standards: 2.8.11.D Formulate 2.8.11.D Formulate inequalities to model routine and non-inequalities to model routine and non-

routine problems.routine problems.

An An inequalityinequality is a is a mathematical statement mathematical statement involving <, >, involving <, >, >>, , <<, or , or ..

Properties of InequalitiesProperties of InequalitiesFor all real numbers For all real numbers a, b, a, b, and and cc, where , where a a << b:b:

Addition PropertyAddition Property a + c a + c << b + c. b + c.

Subtraction PropertySubtraction Property a – c a – c << b – c. b – c.

Multiplication PropertyMultiplication Property Beginning with a ≤ c Beginning with a ≤ c If c is positive, then ac If c is positive, then ac << bc. bc. If c is negative, then ac If c is negative, then ac >> bc. bc.

Division PropertyDivision Property Beginning with a ≤ c Beginning with a ≤ c If c is positive, then a If c is positive, then a c c << b b c. c. If c is If c is negative, then a negative, then a c c >> b b c. c.

Any value of a variable that makes an inequality true Any value of a variable that makes an inequality true is a is a solution of the inequality.solution of the inequality.

II. Solve each inequality and graph the II. Solve each inequality and graph the solution on the number line.solution on the number line.

If the inequality symbol opens towards the If the inequality symbol opens towards the variable then shade to the variable then shade to the rightright on the # on the # line. (Example: x>2 or 5<x)line. (Example: x>2 or 5<x)

If the inequality symbol opens away from If the inequality symbol opens away from the variable then shade to the the variable then shade to the leftleft on the # on the # line. (Example: x<4 or -3>x)line. (Example: x<4 or -3>x)

If If >> or or << , then shade in the circle. If >, < , then shade in the circle. If >, <

or or then leave the circle open. then leave the circle open.

II. Solve each inequality and graph the II. Solve each inequality and graph the solution on the number line.solution on the number line.

Ex 1. Ex 1. 4x – 5 4x – 5 >> 13 13

Ex 2. Ex 2. 4 – 3p > 16 – p4 – 3p > 16 – p

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

Ex. 4Ex. 4Claire’s test average in her world history class is 90. Claire’s test average in her world history class is 90. The test average is 2/3 of the final grade and the The test average is 2/3 of the final grade and the homework is 1/3 of the final grade. What homework homework is 1/3 of the final grade. What homework average does Claire need in order to average does Claire need in order to

have a final grade of at least a 93%?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) > 33H > 99

III. Compound Inequalities – is a pair of III. Compound Inequalities – is a pair of inequalities joined by inequalities joined by andand or or or.or.

To solve an inequality To solve an inequality involving involving ANDAND, find the , find the values of the variable values of the variable that satisfy that satisfy bothboth inequalities. An inequalities. An ANDAND compound inequality compound inequality either has an answer either has an answer because the inequalities because the inequalities INTERSECTINTERSECT or a or a nono solutionsolution answer, answer, because the inequalities because the inequalities DON’TDON’T INTERSECTINTERSECT..

To solve an inequality To solve an inequality involving involving OROR, find those , find those values of the variable that values of the variable that satisfy satisfy at least oneat least one of of inequalities. An inequalities. An OROR compound inequality either compound inequality either has an inequality solution has an inequality solution because the inequalities because the inequalities DON’T INTERSECT or all DON’T INTERSECT or all real numbersreal numbers because the because the inequalities inequalities INTERSECTINTERSECT and and COVERCOVER THE ENTIRE THE ENTIRE NUMBER LINENUMBER LINE..

III. Compound InequalitiesIII. Compound Inequalities

Graph the solution of each compound Graph the solution of each compound inequality on a number line.inequality on a number line.

Ex 1. Ex 1. 2x + 1 2x + 1 >> 3 3 and and 3x – 4 3x – 4 << 17 17


Graph the solution of each compound inequality on Graph the solution of each compound inequality on a number line.a number line.

Ex 2. Ex 2. 2b – 3 2b – 3 >> 1 1 andand 3b + 7 3b + 7 << 1. 1.



Ex 3. Ex 3. 5x + 1 > 21 5x + 1 > 21 oror 3x + 2 < -1 3x + 2 < -1



Ex 4. x + 7 > 4 Ex 4. x + 7 > 4 oror x – 2 < 2. x – 2 < 2.

Writing Activities: Solving InequalitiesWriting Activities: Solving Inequalities

11). Which Properties of Inequality differ from the 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 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.13). Describe two kinds of compound inequalities.

HomeworkHomework

Integrated Algebra II- Section 1.7 Level AIntegrated Algebra II- Section 1.7 Level A

Academic Algebra II- Section 1.7 Level BAcademic Algebra II- Section 1.7 Level B

1.7 introduction to solving inequalities objectives: write, solve, and graph linear inequalities in...

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