6-6 Solving Inequalities Involving Absolute Value Algebra 1 Glencoe McGraw-HillLinda Stamper

GO LA

Which type of compound inequality, “AND” or “OR”, has a greater number of solutions? Why? “Or” because it graphs as opposite rays that continue to infinity. An “AND” type of compound inequality will graph as.... An “OR” type of compound inequality will graph as....

 An absolute-value inequality is an inequality that has one of these forms: When the absolute value is on the left, the Less than symbol represents the “AND” type of inequality. It graphs as a line segment and has less – fewer solutions. When the absolute value is on the left, the Greater than symbol represents the “OR” type of inequality. It graphs as two opposite rays and has a greater number of solutions. I should copy the above notes in my notebook!

Identify the inequality as an “AND” type or “OR” type. Then identify the graph as a line segment or opposite rays. Inequality TypeGraph AND line segment OR opposite rays OR opposite rays AND line segment GO LA GO BEFORE you identify the type of compound inequality you must isolate the absolute value.

 An absolute-value inequality is an inequality that has one of these forms: “AND” type “OR” type To solve an absolute-value inequality, write the two related inequalities – a positive inequality and a negative inequality. When you write the related inequality for the negative value, reverse the inequality symbol.

Do not identify the type of inequality until the absolute value is isolated! Solve. Then graph the solution. Write the positive related inequality. Write the inequality. Write the negative related inequality; LA Isolate the absolute value on one side of the inequality sign. –7 –5 Solve each inequality. O O Write as a single inequality. Graph. and < /

Do not identify the type of inequality until the absolute value is isolated! Solve. Then graph the solution. Write the positive related inequality. Write the inequality. Write the negative related inequality; LA Isolate the absolute value on one side of the inequality sign. –7 –5 Solve. O O Graph. I should copy the above notes in my notebook! < /

Solve. Then graph the solution. Write the positive related inequality. Write the inequality. Write the negative related inequality; GO Isolate the absolute value on one side of the inequality sign. –7 –5 Solve each inequality. O O Reposition. Graph. or > /

Solve. Then graph the solution. Example 1 Example 2

Solve. Then graph the solution. –4 4 Note: Less than symbol represents the “AND” type of inequality and will graph as a line segment. LA Example 1 –4 4 LA Example 1

Solve. Then graph the solution. –8 8 GO Example 2 Note: Greater than symbol represents the “OR” type of inequality and will graph as opposite rays.

Solve. Then graph the solution. Absolute value cannot be less than zero (cannot be negative)! Thus there are no values that will be less than negative one. 

Solve. Then graph the solution. Absolute value will be zero or greater (it cannot be negative). Thus x can be any real number and the absolute value will be greater than negative one. all real numbers 0

Solve. Then graph the solution. Example 3 Example 4 Example 6 Example 7 Example 8 Example 5 Example 9

Example 3 Solve. Then graph the solution Note: Greater than symbol represents the “OR” type of inequality and will graph as opposite rays. GO

Example 4 Solve. Then graph the solution. –1 7 O O Isolate the absolute value on one side of the inequality sign. GO

Example 5 Solve. Then graph the solution. Absolute value cannot be less than zero (cannot be negative)! Thus there are no values that will be less than negative one. 

< Example 6 Solve. Then graph the solution. / –5 –1 LA

Example 7 Solve. Then graph the solution. 2 LA

Example 8 Solve. Then graph the solution. Absolute value will be zero or greater (it cannot be negative). Thus x can be any real number and the absolute value will be greater than negative one. all real numbers 0

Example 9 Solve and then graph. GO 12

When solving an absolute-value inequality: The less than symbol represents the “AND” type of inequality. It graphs as a line segment and has less (fewer) solutions. The greater than symbol represents the “OR” type of inequality. It graphs as two opposite rays and has a greater number of solutions. GO LA

6-A12 Pages # 8–16,23–26,46-51.