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Find algebraically the solution set of the equation four 𝑥 multiplied by the absolute value of 𝑥 minus four 𝑥 equals zero.
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So with this kind of problem, we have two situations to consider.
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And that’s because we’re looking at the absolute value of 𝑥.
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So we’re only wanting the positive value of that 𝑥.
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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.
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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.
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So what we’re gonna do is start the situation where we go down the left-hand path.
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So here we’re looking at values where 𝑥 is greater than or equal to zero, so we can just consider our 𝑥.
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So, four 𝑥 multiplied by 𝑥 minus four 𝑥 equals zero.
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So, if we multiply it, then we’re gonna get four 𝑥 squared minus four 𝑥 equals zero.
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So next, what we can do is divide through by four, so we’re gonna get 𝑥 squared minus 𝑥 equals zero.
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So now if we factor this, we get 𝑥 multiplied by 𝑥 minus one equals zero.
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So therefore, we can say that the solution to this equation is that 𝑥 is equal to zero or one.
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And that’s because if we had 𝑥 equals zero, then zero multiplied by anything is equal to zero.
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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.
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Okay, great.
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Now let’s take a look at the path down the right-hand side.
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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.
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So we’re gonna have four 𝑥 multiplied by negative 𝑥 minus four 𝑥 equals zero.
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So that’s gonna give us negative four 𝑥 squared minus four 𝑥 equals zero.
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So then, if we divide through by negative four, we’ll get 𝑥 squared plus 𝑥 equals zero.
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So then we can factor this to get the solutions.
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So we’re gonna have 𝑥 multiplied by 𝑥 plus one equals zero.
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So therefore, 𝑥 is gonna be equal to zero or negative one.
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And we get those results because if 𝑥 is equal to zero, then zero multiplied by anything is zero.
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And if 𝑥 was equal to negative one, then negative one plus one is zero and again multiplied by anything is zero.
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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.