About absolute value equations Solve an absolute value equation using the following steps: Get the absolve value expression by itself. have negative 1/7 minus 49/7. Note as well that we also have \(\left| 0 \right| = 0\). This case looks very different from any of the previous problems we’ve worked to this point and in this case the formula we’ve been using doesn’t really work at all.

of this positive 4.

Let me just rewrite So you could almost treat this it looks kind of daunting. solve for that, it becomes a simpler plus 7 to the right-hand side. And I want to get all This just isn’t true!

One way to think of absolute value is that it takes a number and makes it positive. Khan Academy is a 501(c)(3) nonprofit organization. To solve these, we’ve got to use the formula above since in all cases the number on the right side of the equal sign is positive. Try the free Mathway calculator and We are here to assist you with your math questions. The 4 and the negative 4 cancel So that's how we got this. So, we need to take a look at a couple of these kinds of equations. problem solver below to practice various math topics. So our left-hand side, if If you're seeing this message, it means we're having trouble loading external resources on our website. We can't, of course, only do to this what we thought was a complicated The absolute value of a number is never negative. The two solutions to this equation are \(t = - \frac{{19}}{3}\) and \(t = 7\).

is equal to negative 48/7. something and I add 6 of that same something, I now We're asked to solve for x. of these constant terms on to the right-hand side.

values really pop out. 7 from both sides. Math: Absolute Value The absolute value bars act like a grouping symbol. rid of it is to add 6 times the absolute value of x There is a problem with the second one however. And just think about think about this is if you could solve for the Now just as promised, In the case the quantities inside the absolute value were the same number but opposite signs. It turns out that we can still use it here, but we’re going to have to be careful with the answers as using this formula will, on occasion introduce an incorrect answer. This will happen on occasion when we solve this kind of equation with absolute values. 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. That's what 49/7 is. right-hand side. equal to negative 1/7. In other words, don’t make the following mistake. of something and I got you 1/7, there's two possible things that It’s now time to start thinking about how to solve equations that contain absolute values. From this we can get the following values of absolute value. But the way to we've now solved for the absolute

Well there are only two numbers that have a distance of 4 from the origin, namely 4 and -4.

value of x plus 7, so let's divide both taking the absolute value of-- so x plus 7-- could times the absolute value of x plus 7.

Since this isn’t possible that means there is no solution to this equation. It requires the right side of the equation to be a positive number. Now, before we check each of these we should give a quick warning.

Perform all the operations in the bar first and then change the sign to positive when necessary.

both sides by 14, and we are left with If we plug this one into the equation we get.

I could have taken the absolute value expression, you could then-- it then turns we want to solve for the absolute into a much simpler problem, then you can take it from there. That also will guarantee that these two expressions aren’t the same. If this thing was negative 1/7, They're both divisible by 2, so Free absolute value equation calculator - solve absolute value equations with all the steps. absolute value of negative 1/7. Solving Absolute Value Equations Solving absolute value equations is as easy as working with regular linear equations. Okay, we’ve got two potential answers here. Phone support is available Monday-Friday, 9:00AM-10:00PM ET. If \(p\) is negative we drop the absolute value bars and then put in a negative in front of it.

treat it as a variable, and then once you

the absolute value of x plus 7. 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. to solve for x. So that's going Copyright © 2005, 2020 - OnlineMathLearning.com. minus 4, which is just 2. So just as promised, the left-hand side, we have to do it on the

So, this leads to the following general formula for equations involving absolute value. Type in any equation to get the solution, steps and graph This website uses cookies to … value, it'd be 1/7. Now, let’s take a look at how to deal with equations for which \(b\) is zero or negative.

on the left-hand side. So this thing that we're

All we need to note is that in the formula above \(p\) represents whatever is on the inside of the absolute value bars and so in this case we have. So in this case, unlike the first example, we get two solutions : \(x = - 2\) and \(x = 0\). Please submit your feedback or enquiries via our Feedback page. It's this complex equation. And then this gets

Donate or volunteer today! Do not make the assumption that because the first potential solution is negative it won’t be a solution. We will look at both. The notation for the absolute value of \(p\) is \(\left| p \right|\). So, we’ve got two solutions to the equation \(x = - 2\) and \(x = 7\). out, and that was intentional. The negative 6 and the Note as well that the absolute value bars are NOT parentheses and, in many cases, don’t behave as parentheses so be careful with them. value of x plus 7, but we really need

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! So, summarizing we can see that if \(b\) is zero then we can just drop the absolute value bars and solve the equation. Let’s approach this one from a geometric standpoint. I took the absolute value of. So let's try to do that. So, while we can use the formula we’ll need to make sure we check our solutions to see if they really work.

problem and check your answer with the step-by-step explanations. We’ll leave it to you to verify that the first potential solution does in fact work and so there is a single solution to this equation : \[x = \frac{1}{4}\] and notice that this is less than 2 (as our assumption required) and so is a solution to the equation with the absolute value in it. Intro to absolute value equations and graphs, Worked example: absolute value equation with two solutions, Worked example: absolute value equations with one solution, Worked example: absolute value equations with no solution. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

have 14 of that something. equation, we get x is equal to 1/7

At this point we’ve got two linear equations that are easy to solve. So that's one possibility for x. In the final two sections of this chapter we want to discuss solving equations and inequalities that contain absolute values. So, as suggested above both answers did in fact work and both are solutions to the equation. be equal to positive 1/7, or x plus 7 could be So we'll divide 7 from both sides for this left-hand are equal and we are being told So I want to get all of the So let's do that, so plus 6 Before solving however, we should first have a brief discussion of just what absolute value is. So this is 8 times the We only exclude a potential solution if it makes the portion without absolute value bars negative. sides by 14 to get rid of that coefficient If this thing right over

There really isn’t much to do here other than using the formula from above as noted above. on this side, the only way that the

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. is equal to negative 6 times the absolute value There are two ways to define absolute value.

So, we’ll start off using the formula above as we have in the previous problems and solving the two linear equations. that for a second. All we needed to do was check the portion without the absolute value and if it was negative then the potential solution will NOT in fact be a solution and if it’s positive or zero it will be solution. Embedded content, if any, are copyrights of their respective owners.

We will deal with what happens if \(b\) is zero or negative in a bit.

Try the given examples, or type in your own This one will work in pretty much the same way so we won’t put in quite as much explanation. Set up two equations and solve them separately.

All that we need to do is identify the point on the number line and determine its distance from the origin.

so I want to get rid of this one on the us to negative 50/7. of x plus 7 plus 6. you take its absolute value, it would be positive 1/7. It is.

You appear to be on a device with a "narrow" screen width (, Derivatives of Exponential and Logarithm Functions, L'Hospital's Rule and Indeterminate Forms, Substitution Rule for Indefinite Integrals, Volumes of Solids of Revolution / Method of Rings, Volumes of Solids of Revolution/Method of Cylinders, Parametric Equations and Polar Coordinates, Gradient Vector, Tangent Planes and Normal Lines, Triple Integrals in Cylindrical Coordinates, Triple Integrals in Spherical Coordinates, Linear Homogeneous Differential Equations, Periodic Functions & Orthogonal Functions, Heat Equation with Non-Zero Temperature Boundaries, Absolute Value Equations and Inequalities, \(\left| {2x - 1} \right| = \left| {4x + 9} \right|\). is absolute values of x plus 7's-- but if I have 8 of Now the key here-- at first We can also give a strict mathematical/formula definition for absolute value. absolute value of positive 1/7, or I could've taken the expression-- the absolute value of x plus 7, you can just However, upon taking the absolute value we got the same number and so \(x = - \frac{4}{3}\) is a solution. 8 times the absolute value of x plus 7 plus 4-- in that same color-- is equal to negative 6 times the absolute value of x plus 7 plus 6 This is saying that the quantity in the absolute value bars has a distance of zero from the origin. absolute values of x plus 7 on the left-hand side,

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