Warm-up – pick up handout up front Solve by factoring. 1000x 3 -10x Answers: 1.x=0, x=1/10, x= -1/10 HW 1.7A (2-14 evens, 21-24, ) Solve for x

HW 1.6B (31-39 all) HW 1.6C (61-75 odds)

Lesson 1.7A Solving linear inequalities and the types of notation Objective: To be able to use interval notation when solving linear inequalities, recognize inequalities with no solution or all real numbers as a solution. The set of all solutions is called the solution set of the inequality. Set-builder notation and a new notation called interval notation, are used to represent solution sets. (See Handout!)

Interval Notation: See Handout

Example 1: (-1,4] Graph and write the solution in set-builder notation. You Try: (-∞, -4 ) (-1,4] = {x -1 < x ≤ 4} ) -4 0 ( ] (-∞, -4) = { x x < -4} Interval Notation Set Builder Notation The answer is read x such that -1 is less than x which is less than or equal to 4.

Graphing intervals on a number line! Example 2: Graph each interval on a number line. {x 2 < x < 3} ( ] 2 3 (2,3] Write answer using set builder notation.

You try: Graph and write in set builder notation. {x 1 < x < 6} (-3,7] {x -3 < x < 7} ( ] -3 7 [ ) 1 6 [1,6)

Example 3: Solve and graph this linear inequality. -2x - 4 > x + 5 x < -3 ) -3 0 Remember to switch the sign when you multiply or divide by a negative.

You Try!! 3x+1 > 7x – 15 Use interval notation to express the solution set. Graph the solution. Answer: (-∞, 4) ) 4

Inequalities with Unusual Solution Sets Some inequalities have no solution. Example 4: x > x+1 There is no number that is greater than itself plus one. The solution set is an empty set ( this is a zero with a slash through it)

Notes Continued: Like wise some inequalities are true for all real numbers such as: Example 5: x<x+1. Every real number is less than itself plus 1. The solution set is { x x is a real number} Interval notation, or all real numbers.

Notes Continued: When solving an inequality with no solution, the variable is eliminated and there will be a false solution such as 0 > 1. When solving an inequality that is all real numbers, the variable is eliminated and there will be a true solution such as 0 < 1.

Example 6: Solving a linear inequality for a solution set. A. 2 (x + 4) > 2x + 3 B. x + 7 < x – 2 The inequality 8>3 is true for all values of x. The solution set is {x x is a real number} or solution: 8 > 3 solution: 7 < -2 The inequality 7 < -2 is false for all values of x. The solution set is

You try: 3(x + 1) > 3x + 2 3x + 3 > 3x > 2 Summary: Describe the ways in which solving a linear inequality is different from solving a linear equation. The solution set is all real numbers. (-∞,∞)