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Notice that this does require the \(b\) be a positive number. Note that we really didn’t need to plug the solution into the whole equation here.

In this definition we are going to think of \(\left| p \right|\) as the distance of \(p\) from the origin on a number line. So let's do that, so plus 6 There are two ways to define absolute value. There is a geometric definition and a mathematical definition.

that to the right-hand side. From this we can get the following values of absolute value. If \(p\) is negative we drop the absolute value bars and then put in a negative in front of it.

So, let’s see a couple of quick examples. 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! solve for that, it becomes a simpler

So, we need to take a look at a couple of these kinds of equations. So let's try to do that. In the case the quantities inside the absolute value were the same number but opposite signs. absolute value problem. values really pop out. equation are negative 48/7 and negative 50/7. problem and check your answer with the step-by-step explanations. So, as we’ve seen in the previous set of examples we need to be a little careful if there are variables on both sides of the equal sign. I could have taken the us to negative 50/7. is absolute values of x plus 7's-- but if I have 8 of So this is 8 times the 4 right over there, but if we do it on I took the absolute value of. The two solutions to this equation are \(t = - \frac{{19}}{3}\) and \(t = 7\). treat it as a variable, and then once you This tells us to look at the sign of \(p\) and if it’s positive we just drop the absolute value bar. of these constant terms on to the right-hand side.

minus 4, which is just 2. here were equal to 1/7, you take its absolute We can also give a strict mathematical/formula definition for absolute value. absolute value of x plus 7. For example, if \(x = - 1\) we would get.

Set up two equations and solve them separately. At this point we’ve got two linear equations that are easy to solve.

In other words, don’t make the following mistake. equation, we get x is equal to 1/7 So we'll divide That also will guarantee that these two expressions aren’t the same. And I want to get all Now, we won’t need to verify our solutions here as we did in the previous two parts of this problem. 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. plus 7 to the right-hand side. If these two things 7 from both sides for this left-hand Math: Absolute Value The absolute value bars act like a grouping symbol. 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. Likewise, if \(b\) is negative then there will be no solution to the equation. Note as well that we also have \(\left| 0 \right| = 0\). If there is a negative outside the absolute value bar, it stays there. 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. Let’s approach this one from a geometric standpoint. However, the argument of the previous absolute-value expression was x + 2.In this case, only the x is inside the absolute-value bars. Now, if you think about it we can do this for any positive number, not just 4.

equal to negative 1/7. This is saying that the quantity in the absolute value bars has a distance of zero from the origin. So, as suggested above both answers did in fact work and both are solutions to the equation.

Donate or volunteer today! About absolute value equations Solve an absolute value equation using the following steps: Get the absolve value expression by itself. on this side, the only way that the of something and I got you 1/7, there's two possible things that If this thing right over In the final two sections of this chapter we want to discuss solving equations and inequalities that contain absolute values. Also, we will always use a positive value for distance. reason through this? taking the absolute value of-- so x plus 7-- could is equal to negative 6 times the absolute value Now, let’s take a look at how to deal with equations for which \(b\) is zero or negative. are equal and we are being told If we plug this one into the equation we get. possibility we would get x is equal to-- so we Knowing that the contents within the absolute value symbol should equal 4, it follows that: 3 x + 9 = 6 and 3 x +9 = − 6 ∴ x = − 1, − 5

times the absolute value of x plus 7. Now, remember that absolute value does not just make all minus signs into plus signs!

Likewise, there is no reason to think that we can only have one absolute value in the problem. have 14 of that something.

Easiest way to get We're just going to solve for right-hand side as well. However, once we put variables inside them we’ve got to start being very careful. However, upon taking the absolute value we got the same number and so \(x = - \frac{4}{3}\) is a solution. so I want to get rid of this one on the absolute value of negative 1/7. We get the same number on each side but with opposite signs.

And then the other There is a problem with the second one however.

6 x plus 7's cancel out, or absolute values of rid of it is to add 6 times the absolute value of x They're both divisible by 2, so both sides by 14, and we are left with 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. And just think about

Since this isn’t possible that means there is no solution to this equation. 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, \(\left| {2x - 1} \right| = \left| {4x + 9} \right|\). The notation for the absolute value of \(p\) is \(\left| p \right|\). The absolute value of a number is the distance of the number from zero on the number line. sides by 14 to get rid of that coefficient 14 absolute values of x plus 7, 14 times the

So if I take the absolute value So you could almost treat this So, all together there is a single solution to this equation : \[x = \frac{1}{4}\]. It’s now time to start thinking about how to solve equations that contain absolute values. Solving Absolute Value Equations Solving absolute value equations is as easy as working with regular linear equations. To this point we’ve only looked at equations that involve an absolute value being equal to a number, but there is no reason to think that there has to only be a number on the other side of the equal sign. absolute value expression, you could then-- it then turns the absolute value of x plus 7.

absolute values in it.

Both with be solutions provided we solved the two equations correctly. on the left-hand side. value of x plus 7, but we really need Now, if we think of this from a geometric point of view this means that whatever \(p\) is it must have a distance of 4 from the origin. because we don’t know the value of \(x\). Easiest way is to subtract this equation so that the absolute The only additional key step that you need to remember is to separate the original absolute value equation into two parts: positive and negative (±) components.

The definitions above are easy to apply if all we’ve got are numbers inside the absolute value bars. It is. So I want to get all of the And then this gets value, it'd be 1/7. Now, before we check each of these we should give a quick warning. 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. We will look at equations with absolute value in them in this section and we’ll look at inequalities in the next section. Or in other words, we must have. This will happen on occasion when we solve this kind of equation with absolute values. to solve for x. Free absolute value equation calculator - solve absolute value equations with all the steps. You have these In the case we got the same value inside the absolute value bars. We are here to assist you with your math questions. We will look at both. It's this complex equation. This one is pretty much the same as the previous part so we won’t put as much detail into this one. minus-- and 7 we can rewrite as 49/7, which Note that these give exactly the same value as if we’d used the geometric interpretation. We will deal with what happens if \(b\) is zero or negative in a bit.

So the absolute value of 6 is 6, and the absolute value of −6 is also 6.

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.

If you're seeing this message, it means we're having trouble loading external resources on our website. 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. So this thing that we're However, if we think about this a little we can see that we’ll still do something similar here to get a solution. 7 from both sides. So, we’ll start off using the formula above as we have in the previous problems and solving the two linear equations. So the two solutions You'll see what I mean. So that's going to be Consider the following number line. Let's try to solve absolute values of x plus 7 on the left-hand side, At first glance the formula we used above will do us no good here. Before solving however, we should first have a brief discussion of just what absolute value is.

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. call it, the thing that's multiplying the absolute it looks kind of daunting. absolute value of x plus 7 plus 4-- in that same color-- All that we need to do is identify the point on the number line and determine its distance from the origin. We're just subtracting Khan Academy is a 501(c)(3) nonprofit organization. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. Now just as promised,

value of x plus 7. So I want to get rid have negative 1/7 minus 49/7. First, the numbers are clearly not the same and so that’s all we really need to prove that the two expressions aren’t the same. x plus 7's cancel out, and that was intentional.

into a much simpler problem, then you can take it from there.

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