Aim: How do we solve absolute value inequalities? Do Now: Solve for x |4 – x| = 3 2) |5x + 3| = -2 HW #4 – Chapter 1 pg 16 #21-26

Solving Inequalities To solve an inequality, use the same procedure as solving an equation with one exception. When multiplying or dividing by a negative number, reverse the direction of the inequality sign. -3x < 6 divide both sides by -3 -3x/-3 > 6/-3 x > -2

Solving Inequalities 3b - 2(b - 5) < 2(b + 4) 1

“and’’ Statements (Conjunctions) can be Written in Two Different Ways 2. 8 < m+6 and m+6 < 14 These inequalities can be solved using two methods.

Method One Example : 8 < m + 6 < 14 Rewrite the compound inequality using the word “and”, then solve each inequality. 8 < m + 6 and m + 6 < 14 2 < m m < 8 m >2 and m < 8 2 < m < 8 Graph the solution: 8 2

Method Two Example: 8 < m + 6 < 14 To solve the inequality, isolate the variable by subtracting 6 from all 3 parts. 8 < m + 6 < 14 -6 -6 -6 2 < m < 8 Graph the solution. 8 2

‘or’ Statements (Disjunctions) Example: x - 1 > 2 or x + 3 < -1 x > 3 x < -4 x < -4 or x >3 Graph the solution. 3 -4

Solving Absolute Value Inequalities Solving absolute value inequalities is a combination of solving absolute value equations and inequalities. Step 1: Isolate the absolute value on the left side Step 2: Rewrite the inequality as a conjunction or a disjunction. If you have a you are working with a conjunction or an ‘and’ statement. Remember: “Less thand” If you have a you are working with a disjunction or an ‘or’ statement. Remember: “Greator”

Step 4. Graph on a number line: Step 3. Consider 2 cases: ●For the first case, drop the absolute value bars and solve the inequality. ●For the second case, you have to negate the right side of the inequality and reverse the inequality sign. Step 4. Graph on a number line: ● For conjunctions, , the graph is between the values. ● For disjunctions, , the graph goes in opposite directions of the values.

Example 1: Graph goes in opposite directions |2x + 1| > 7 This is an ‘or’ statement. (Greator). Disjunction First Case – Drop and solve 2nd Case, reverse the inequality sign negate the right side value and solve. . Graph the solution. |2x + 1| > 7 2x + 1 > 7 (Case 1) 2x + 1 <-7 (Case 2) x > 3 or x < -4 3 -4 Graph goes in opposite directions

Example 2: Between the values This is an ‘and’ statement. (Less thand). Conjunction. First Case. Drop and solve. 2nd Case, reverse the inequality sign and negate the right side value. Solve each inequality. Graph the solution. |x -5|< 3 x -5< 3 x -5> -3 x < 8 and x > 2 2 < x < 8 8 2 Between the values

Solve and Graph 1) |y – 3| > 1 2) |p + 2| < 6