The Purpose of Flipping Inequality Symbols when Multiplying by a Negative Number and Solving for Absolute Value Inequalities!

Many of you solved absolute value inequality problems and regular inequality problems, but never understood why the inequality symbols must be flipped. Well...

Absolute Value Inequalities

The definition of absolute value is, the distance a number is from zero. Therefore, a negative number and a positive number have the same distance from zero. For example:

As you notice, -4 and 4 has the same absolute value. When solving for absolute value inequalities the sign is flipped because of the number line. On the number line both negative and positive numbers count because they both have a distance to Zero. 

Regular Inequality Problems:

Inequality problems on a number line show what values the variable is greater or less than. Therefore, when multiplying by a negative number the sign flips because it is actually less than what the variable can be. On a number line, the inequality sign must be flipped to satisfy the inequality. For Example,

As you can see, the flipping of the sign just makes the inequality true because on a number line a negative number is less than the variable or number used in the problem.

 

Question:

Solve for x: -3|x-9|>27. How many times did you flip the inequality sign?

Sources: 

• http://neaportal.k12.ar.us/wp-content/uploads/2010/08/sei2a14_solve-and-graph-absolute-value-equations-and-inequalities.jpg
• https://d3pof0dg6ipqi1.cloudfront.net/learning_preview/10913/image/large_hqdefault.jpg