Solving Linear Inequalities MATH 018 Combined Algebra S. Rook

2 Overview Section 2.7 in the textbook: –Graphing inequalities on a number line & interval notation –Using the Addition Property of Inequality –Using the Multiplication Property of Inequality –Solving inequalities using both properties

3 Graphing Inequalities on a Number Line & Interval Notation

44 Solution Set Most equations have 1 solution whereas most inequalities have an infinite number of solutions Solution set – all values that satisfy an inequality Often a solution set is expressed using a number line

55 Graphing Inequalities on a Number Line Consider the inequality x > 1 –What are some values of x that make the inequality true? –Thus x can be any value greater than 1 Which direction on the number line indicates increasing values? –Can x = 1 be a solution to the inequality? Since x = 1 is not in the solution set, we put a parenthesis around 1 on the number line

66 Graphing Inequalities on a Number Line (Continued) Now consider x ≥ 1 –Graphed ALMOST the same way EXCEPT: Is x = 1 included in the solution set? –Since x = 1 is part of the solution set, we put a bracket around 1 on the number line Given x a: –Parenthesis goes around a on the number line (not inclusive) Given x ≤ a or x ≥ a: –Bracket goes around a on the number line (inclusive)

7 Interval Notation Easy once the graph is obtained Represents the “endpoints” of the graph –First “value” is what is shaded furthest to the left on the graph –Second “value” is what is shaded furthest to the right on the graph –A shaded arrow represents oo Parentheses ALWAYS go around infinity

Graphing Inequalities on a Number Line & Interval Notation (Example) Ex 1: Graph on a number line AND write in interval notation: a) x < 9 ⁄ 2 b) -5 < y ≤ 2 8

9 Addition Property of Inequality

10 Addition Property of Inequality Works the same way as the Addition Property of EQuality When a number is being added or subtracted to the variable: –Add the OPPOSITE number to BOTH SIDES Consider 2 < 7 –What happens when we add 2 to both sides? –What happens when we subtract 5?

Addition Property of Inequality (Example) Ex 2: Solve, graph, AND write the solution set in interval notation: a) x + 3 > 4 b) 3y – 4 ≤ 2y – 9 11

12 Multiplication Property of Inequality

13 Multiplication Property of Inequality ALMOST the same as the Multiplication Property of EQuality When the variable is being multiplied by a number: –Divide BOTH SIDES by the number INCLUDING THE SIGN Consider 4 > 2 –What happens when we divide by 2? –What happens when we divide by -2?

14 Multiplication Property of Inequality (Continued) Thus, when we DIVIDE an INEQUALITY by a NEGATIVE number: –Switch the direction of the inequality –Failing to SWITCH the inequality when DIVIDING by a NEGATIVE number is a common mistake!

Multiplication Property of Inequality (Example) Ex 3: Solve, graph, AND write the solution set in interval notation: a) 6x ≥ -18 b) -8y < 14 c) -z ≤ -6 15

16 Solving Inequalities Using Both Properties

17 Solving Inequalities Using Both Properties (Example) Ex 4: Solve, graph, AND write the solution set in interval notation: a) -5x – 9 > -2x + 12 b) 4(y – 5) ≥ 3y – (y + 2)

Solving Inequalities Using Both Properties (Example) Ex 5: Solve, graph, AND write the solution set in interval notation: 18

19 Summary After studying these slides, you should know how to do the following: –Graph an inequality on a number line –Understand the Addition and Multiplication Properties of Inequality –Solve and graph inequalities Additional Practice –See the list of suggested problems for 2.7 Next lesson –Solving Absolute Value Equations (Section E.1)