# Solving absolute value equations and inequalities pdf

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- Teaching Absolute Value Equations and Inequalities
- 1.7 Solving Absolute Value Equations and Inequalities
- solving absolute value equations or inequalities
- Solving Absolute Value Equations and Inequalities (Day 1 of 2)

*Absolute Value Equations and Inequalities Part 1 The concept of absolute value equations and inequalities. Absolute Value Equation Worksheets.*

The purpose of this lesson is for my students to develop a conceptual understanding of what absolute value means, and apply it to solving Absolute Value Equations and Inequalities. So in the Introduction Activity , I have students complete a hands on activity with a table partner. This activity should help students think about the possible solutions to Absolute Value Equations and Inequalities in terms of "distance from zero.

## Teaching Absolute Value Equations and Inequalities

In addition, the absolute value of a real number can be defined algebraically as a piecewise function. Use this theorem to solve absolute value equations algebraically. To visualize these solutions, graph the functions on either side of the equal sign on the same set of coordinate axes. The solutions correspond to the points of intersection. To apply the theorem, the absolute value must be isolated. The general steps for solving absolute value equations are outlined in the following example.

In this case, we can see that the isolated absolute value is equal to a negative number. Recall that the absolute value will always be positive. Therefore, we conclude that there is no solution. Geometrically, there is no point of intersection. In other words, if two absolute value expressions are equal, then the arguments can be the same or opposite. The absolute value of a number represents the distance from the origin. We can graph this solution set by shading all such numbers.

Express this solution set using set notation or interval notation as follows:. In this text, we will choose to express solutions in interval notation. This theorem holds true for strict inequalities as well. In other words, we can convert any absolute value inequality involving " less than " into a compound inequality which can be solved as usual. Here we use open dots to indicate strict inequalities on the graph as follows. To apply the theorem, we must first isolate the absolute value.

Next, apply the theorem and rewrite the absolute value inequality as a compound inequality. Shade the solutions on a number line and present the answer in interval notation. Here we use closed dots to indicate inclusive inequalities on the graph as follows:. Next, we examine the solutions to an inequality that involves " greater than ," as in the following example:. On a graph, we can shade all such numbers.

There are infinitely many solutions that can be expressed using set notation and interval notation as follows:. The theorem holds true for strict inequalities as well. To apply the theorem we must first isolate the absolute value. Shade the solutions on a number line and present the answer using interval notation. Up to this point, the solution sets of linear absolute value inequalities have consisted of a single bounded interval or two unbounded intervals.

This is not always the case. Notice that we have an absolute value greater than a negative number. For any real number x the absolute value of the argument will always be positive.

Hence, any real number will solve this inequality. In this case, we can see that the isolated absolute value is to be less than or equal to a negative number. Again, the absolute value will always be positive; hence, we can conclude that there is no solution. In summary, there are three cases for absolute value equations and inequalities.

Solve and graph the solution set. In addition, give the solution set in interval notation. Learning Objectives Review the definition of absolute value. Solve absolute value equations. Solve absolute value inequalities. Solution Begin by isolating the absolute value.

Case 2: An absolute value inequality involving " less than. Case 3: An absolute value inequality involving " greater than. Absolute value equations can have up to two solutions. Remember to isolate the absolute value before applying these theorems. On one side write the theorem, and on the other write a complete solution to a representative example. Share your strategy for identifying and solving absolute value equations and inequalities on the discussion board. Make your own examples of absolute value equations and inequalities that have no solution, at least one for each case described in this section.

Illustrate your examples with a graph. Answer 1. Answer may vary.

## 1.7 Solving Absolute Value Equations and Inequalities

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## solving absolute value equations or inequalities

Absolute Value Equations and Inequalities are most easily related to Distance. We have an Algebra 2 Lesson that does a good job of explaining this. When it comes to Algebra 1 this is probably one of the toughest lessons for the students all year.

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### Solving Absolute Value Equations and Inequalities (Day 1 of 2)

In addition, the absolute value of a real number can be defined algebraically as a piecewise function. Use this theorem to solve absolute value equations algebraically. To visualize these solutions, graph the functions on either side of the equal sign on the same set of coordinate axes. The solutions correspond to the points of intersection. To apply the theorem, the absolute value must be isolated.

You have found 1 out of 13 Christmas decorations! All are free! Thank you for using the timer! We noticed you are actually not timing your practice. There are many benefits to timing your practice , including:. You do not have the required permissions to view the files attached to this post. Watch this Video.

Equivalencies Used to Solve Absolute Value Equations. & Inequalities by Chad Mattingly. When solving equations and inequalities involving absolute values.