Absolute value equations

Join million happy users! Sign Up free of charge:. Join with Office Join with Facebook. Create my account. Transaction Failed! Please try again using a different payment method.  We want your feedback optional. Cancel Send. Generating PDF See All implicit derivative derivative domain extreme points critical points inverse laplace inflection points partial fractions asymptotes laplace eigenvector eigenvalue taylor area intercepts range vertex factor expand slope turning points.Simpler Harder Special Case. When we take the absolute value of a number, we always end up with a positive number or zero. Whether the input was positive or negative or zerothe output is always positive or zero.

This property — that both the positive and the negative become positive — makes solving absolute-value equations a little tricky. But once you learn the "trick", they're not so bad. Let's start with something simple:. Solving Absolute Value Equations. So then x must be equal to 3 or equal to —3. But how am I supposed to solve this if I don't already know the answer?

The "minus" sign in " — x " just indicates that I am changing the sign on x. It does not indicate a negative number. This distinction is crucial! Whatever the value of x might be, taking the absolute value of x makes it positive. Since x might originally have been positive and might originally have been negative, I must acknowledge this fact when I remove the absolute-value bars.

I do this by splitting the equation into two cases. For this exercise, these cases are as follows:. We can, by the way, verify the above solution graphically. For instance:. If you're wanting to check your answers on a test before you hand it init can be helpful to plug each side of the original absolute-value equation into your calculator as their own functions; then ask the calculator for the intersection points. Of course, any solution can also be verified by plugging it back into the original exercise, and confirming that the left-hand side LHS of the equation simplifies to the same value as does the right-hand side RHS of the equation.

Graphing Absolute-Value Functions

For the equation above, here's my check:. If you're ever in doubt about your solution to an equation, try graphing or else try plugging your solution back into the original question.

Checking your work is always okay! The step in the above, where the absolute-value equation was restated in two forms, one with a "plus" and one with a "minus", gives us a handy way to simplify things: When we have isolated the absolute value and go to take off the bars, we can split the equation into two cases; we will signify these cases by placing a "minus" on the opposite side of the equation for one case and a "plus" on the opposite side for the other.

Here's how this works:. The absolute value is isolated on the left-hand side of the equation, so it's already set up for me to split the equation into two cases. To clear the absolute-value bars, I must split the equation into its two possible two cases, one each for if the contents of the absolute-value bars that is, if the "argument" of the absolute value is negative and if it's non-negative that is, if it's positive or zero. To do this, I create two new equations, where the only difference between then is the sign on the right-hand side.

First, I'll do the "minus" case:. Now I need to check my solutions.

Absolute Value

I'll do this by plugging them back into the original equation, since the grader can't see me checking plots on my graphing calculator. First, I'll isolate the absolute-value part of the equation; that is, I'll get the absolute-value expression by itself on one side of the "equals" sign, with everything else on the other side:.

Now I'll clear the absolute-value bars by splitting the equation into its two cases, one for each sign on the argument. First I'll do the negative case:.

The exercise doesn't tell me to check, so I won't. But, if I'd wanted to, I could have plugged "abs 2X—3 —4" and "3" into my calculator as Y1 and Y2, respectivelyand seen that the intersection points were at my x -values. My answer is:. Page 1 Page 2 Page 3. All right reserved. Web Design by.An absolute value equation is an equation that contains an absolute value expression. The equation. An absolute value equation has no solution if the absolute value expression equals a negative number since an absolute value can never be negative.

When solving an absolute value inequality it's necessary to first isolate the absolute value expression on one side of the inequality before solving the inequality. Share on Facebook. Search Pre-Algebra All courses. All courses. Algebra 1 Discovering expressions, equations and functions Overview Expressions and variables Operations in the right order Composing expressions Composing equations and inequalities Representing functions as rules and graphs.

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An absolute value equation is solved using the same rules as any other algebraic equation; however, this type of equation has two potential results, derived from a positive equation and a negative equation.

To solve absolute value equations, first isolate the absolute value terms by moving anything outside of the vertical bars to the other side of the equation. Next, solve for the positive value of the equation by isolating the variable. Since the absolute variable can represent 2 numbers, then solve for the negative value by putting a negative sign outside the vertical bars.

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Learn more Explore this Article Setting up the Problem. Calculating the Values. Check Your Work. Tips and Warnings. Related Articles. Article Summary. Part 1 of Straight Lines Curvy Lines. Taking the absolute value of a negative number makes it positive. For this reason, graphs of absolute value functions tend not to look quite like the graphs of linear functions that you've already studied.

However, because of how absolute values behave, it is important to include negative inputs in your T-chart when graphing absolute-value functions. If you do not pick x -values that will put negatives inside the absolute value, you will usually mislead yourself as to what the graph looks like. One of the other students does what is commonly done: he picks only positive x -values for his T-chart:. These points are fine, as far as they go, but they aren't enough; they don't give an accurate idea of what the graph should look like.

In particular, they don't include any "minus" inputs, so it's easy to forget that those absolute-value bars mean something. As a result, the student forgets to take account of those bars, and draws an erroneous graph:. But you're more careful. You remember that absolute-value graphs involve absolute values, and that absolute values affect "minus" inputs. So you pick x -values that put a "minus" inside the absolute value, and you choose quite a few more points. Your T-chart looks more like this:.

While absolute-value graphs tend to look like the one above, with an "elbow" in the middle, this is not always the case. However, if you see a graph with an elbow like this, you should expect that the graph's equation probably involves an absolute value. In all cases, you should take care that you pick a good range of x -values, because three x -values right next to each other will almost certainly not give you anywhere near enough information to draw a valid picture. Note: The absolute-value bars make the entered values evaluate to being always non-negative that is, positive or zero.

As a result, the "V" in the above graph occurred where the sign on the inside was zero. When x equalled —2then the argument that is, the expression inside the bars equalled zero. For all x -values to the right of —2the argument was positive, so the absolute-value bars didn't change anything.Solving absolute value equations is as easy as working with regular linear equations.

The absolute value of any number is either positive or zero. But this equation suggests that there is a number that its absolute value is negative. Although the right side of the equation is negative, the absolute value expression itself must be positive. But it is not, right? Divide both sides of the equation by this value to get rid of the negative sign. Since the absolute value expression and the number are both positive, we can now apply the procedure to break it down into two equations.

You may verify our answers by substituting them back to the original equation. This problem is getting interesting since the expression inside the absolute value symbol is no longer just a single variable. Just be careful when you break up the given absolute value equation into two simpler linear equations, then proceed how you usually solve equations.

You may think that this problem is complex because of the —2 next to the variable x. We have the absolute value symbol isolated on one side and a positive number on the other. Solving this is just like another day in the park! The absolute value expression is not isolated yet. Solving Absolute Value Equations Solving absolute value equations is as easy as working with regular linear equations.

Download Version 1. Download Version 2. We use cookies to give you the best experience on our website. Otherwise, check your browser settings to turn cookies off or discontinue using the site. Cookie Policy.This equation is asking us to find all numbers, xthat are 3 units from zero on the number line. Step 1. If the absolute values of two expressions are equal, then either the two expressions are equal, or they are opposites.

This absolute value equation is set equal to minus 8, a negative number. By definition, the absolute value of an expression can never be negative.

Solving absolute value equations and inequalities

Hence, this equation has no solution. The symbol for no solution is a zero with a slash through it. The correct response: d. Rewrite as two equations and solve for x. Skip to content. Absolute value equations are useful in determining distance and error measurements. We must consider numbers both to the right and to the left of zero on the number line.

Notice that both 3 and -3 are three units from zero. Example 1 suggests a rule that we can use when solving absolute value equations. Solve each equation.

Check the solutions. The absolute value of a number is never negative. This equation has no solution. Break the equation up into two equivalent equations.