How do you write absolute value equations problems

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How do you write absolute value equations problems

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Absolute Value Equations In the final two sections of this chapter we want to discuss solving equations and inequalities that contain absolute values. Before solving however, we should first have a brief discussion of just what absolute value is.


There are two ways to define absolute value. There is a geometric definition and a mathematical definition. We will look at both. Also, we will always use a positive value for distance. Consider the following number line.

From this we can get the following values of absolute value. One way to think of absolute value is that it takes a number and makes it positive.

We do need to be careful however to not misuse either of these definitions. There is also the fact however that the right number is negative and we will never get a negative value out of an absolute value! Well there are only two numbers that have a distance of 4 from the origin, namely 4 and So, this leads to the following general formula for equations involving absolute value.

how do you write absolute value equations problems

Example 1 Solve each of the following. Example 2 Solve each of the following. This is saying that the quantity in the absolute value bars has a distance of zero from the origin. There is only one number that has the property and that is zero itself. That is exactly what this equation is saying however.

Likewise, there is no reason to think that we can only have one absolute value in the problem. So, we need to take a look at a couple of these kinds of equations. Example 3 Solve each of the following. It requires the right side of the equation to be a positive number. There is a problem with the second one however.

This will happen on occasion when we solve this kind of equation with absolute values. So, all together there is a single solution to this equation: We only exclude a potential solution if it makes the portion without absolute value bars negative.

In this case both potential solutions will make the portion without absolute value bars positive and so both are in fact solutions. So in this case, unlike the first example, we get two solutions: Both sides of the equation contain absolute values and so the only way the two sides are equal will be if the two quantities inside the absolute value bars are equal or equal but with opposite signs.


Both with be solutions provided we solved the two equations correctly. However, it will probably be a good idea to verify them anyway just to show that the solution technique we used here really did work properly.

So, as suggested above both answers did in fact work and both are solutions to the equation. If one side does not contain an absolute value then we need to look at the two potential answers and make sure that each is in fact a solution.Search the world's information, including webpages, images, videos and more.

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The absolute number of a number a is written as $$\left | a \right |$$ And represents the distance between a and 0 on a number line.

how do you write absolute value equations problems

An absolute value equation is an equation that contains an absolute value expression. The equation $$\left | x \right |=a$$ Has two solutions x = a and x = -a because both numbers are at the distance a from 0.

Set Up Two Equations

Section Solving Exponential Equations. Now that we’ve seen the definitions of exponential and logarithm functions we need to start thinking about how to solve equations involving them.

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Absolute value inequalities word problem (video) | Khan Academy