Solving Absolute Value Inequalities
Description/Explanation/Highlights
Video Description
In this video we learn to solve absolute value inequalities.
Steps and Key Points to Remember
To solve an absolute value inequality, follow these steps:
- Begin by isolating the part of the inequality that is inside the absolute value sign by performing the same Algebra operations to both sides of the inequality to get rid of everything but the absolute value on one side of the inequality. Remember in an inequality if you multiply or divide by a negative number you must flip the inequality sign.
- Next create two cases to represent positive and negative results.
- If the inequality is > or ≥ then the two cases will become a compound inequality using OR and if the inequality is < or ≤ the cases will be a compound “AND.”
- For > or ≥, rewrite the isolated inequality by first writing the inequality as it appears but without the absolute value signs then write it again leaving the part inside the absolute value signs the same but flip the inequaltiy sign and make all terms on the other side the opposite sign. Separate the two cases with the word “OR”. So |2X+3|>3 would become 2X+3>3 OR 2X+3<−3.
- For < or ≤, rewrite the isolated inequality by first writing the the part inside the absolute value signs as it appears but without the absolute value sign. Then write the negative version of the other side to the left and the other side exactly as it appears to the right. Separate the middle part from the outsides with < or ≤ signs on each side. |−3x+2|< 11 would become −11<−3X+2< 11. Notice the word “AND” does not appear.
- Solve the two sides of the OR inequality separately and put OR between the answers.
- Solve the AND inequality by getting X by itself in the middle of the inequality. This can be accomplished by performing the same Algebra operations to all three parts of the inequality.
- Most inequalities are written with the smallest number first. For example X>4 OR X<−3 could be rewritten as X<−3 OR X>4. 4 > X > −3 could be rewritten by flipping the inequality to make −3 < X < 4 as a final answer.
- Answers can be expressed graphically by graphing on a number line if desired.
Here are some key points to keep in mind when solving absolute value inequalities:
- If you are not familar with solving compound inequalities, review that process first. There is a lesson on compound inequalities on this site.
- Remember > or ≥ become “OR” compound inequalities and < or ≤ become “AND” compound inequalities.
- When isolating the absolute value or solving the inequalities using Algebra operations, don’t forget to flip the inequality signs anytime you divide or multiply by a negative number. i.e. “<” becomes “>”.
- It sometimes helps visually to graph the solution on a number line.
Video Highlights
- 00:00 Introduction to absolute value inequalities
- 00:30 |2X+2|>3 example rewritten with OR
- 01:12 Rewriting the first example into two cases using OR
- 01:47 Solving the two cases with OR
- 03:00 Graphing the solution on a number line
- 04:07 |−3X+2|−8<3 example that requires isolating first
- 04:40 Rewriting the second example into an “AND” compound inequality
- 05:18 Solving the “AND” inequality
- 06:30 |−4X−8|≥16 example rewritten with OR
- 06:40 Rewriting the third example using OR
- 06:58 Solving the compound OR inequality
- 07:45 Rewriting the final answer in correct form with smallest number first
- 08:52 Conclusion and website information