Question Video: Finding the Solution Set of Quadratic Equations Involving Absolute Value Mathematics

Find algebraically the solution set of the equation 4𝑥|𝑥| − 4𝑥 = 0.

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Video Transcript

Find algebraically the solution set of the equation four 𝑥 multiplied by the absolute value of 𝑥 minus four 𝑥 equals zero.

So with this kind of problem, we have two situations to consider. And that’s because we’re looking at the absolute value of 𝑥. So we’re only wanting the positive value of that 𝑥. So first of all, if we consider if 𝑥 is greater than or equal to zero, then it’s going to be positive, so we won’t have to worry about changing or doing anything to our 𝑥. However, if we got 𝑥 is less than zero, then what we’re gonna have to do is have the negative of that to give us a positive. So what we’re gonna do is start the situation where we go down the left-hand path.

So here we’re looking at values where 𝑥 is greater than or equal to zero, so we can just consider our 𝑥. So, four 𝑥 multiplied by 𝑥 minus four 𝑥 equals zero. So, if we multiply it, then we’re gonna get four 𝑥 squared minus four 𝑥 equals zero. So next, what we can do is divide through by four, so we’re gonna get 𝑥 squared minus 𝑥 equals zero. So now if we factor this, we get 𝑥 multiplied by 𝑥 minus one equals zero. So therefore, we can say that the solution to this equation is that 𝑥 is equal to zero or one. And that’s because if we had 𝑥 equals zero, then zero multiplied by anything is equal to zero. And if 𝑥 was equal to one, in the parentheses we’d have one minus one, which would be zero, again, giving us a result of zero.

Okay, great. Now let’s take a look at the path down the right-hand side. So, as we already stated, if 𝑥 is less than zero and we want the result to be positive, then we’re gonna have to have the negative of this. So we’re gonna have four 𝑥 multiplied by negative 𝑥 minus four 𝑥 equals zero. So that’s gonna give us negative four 𝑥 squared minus four 𝑥 equals zero. So then, if we divide through by negative four, we’ll get 𝑥 squared plus 𝑥 equals zero. So then we can factor this to get the solutions. So we’re gonna have 𝑥 multiplied by 𝑥 plus one equals zero.

So therefore, 𝑥 is gonna be equal to zero or negative one. And we get those results because if 𝑥 is equal to zero, then zero multiplied by anything is zero. And if 𝑥 was equal to negative one, then negative one plus one is zero and again multiplied by anything is zero. So therefore, if we bring this all together, the solution set of the equation four 𝑥 multiplied by the modulus or absolute value of 𝑥 minus four 𝑥 equals zero is negative one, zero, and one.

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