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Find algebraically the solution set of the equation the absolute value or modulus of 𝑥 plus four equals 𝑥 plus four.
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So, in this problem, in fact, what we have are two different situations to consider.
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So the reason we’re looking at two different situations to consider is because we’ve got the absolute value or modulus of 𝑥 plus four.
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So actually, what we’re looking at are the positive values of 𝑥 plus four.
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So first of all, let’s consider 𝑥 plus four is greater than or equal to zero.
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We can see that this is already going to be positive, so we’re gonna have to do nothing to this.
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However, if 𝑥 plus four is less than zero, then we’d have to take the negative of that value to make it a positive.
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And that’s what the absolute value function itself is actually doing.
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Now, if we solve for both of our inequalities, on the left-hand side, we’d have 𝑥 is greater than or equal to negative four.
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And we get that by subtracting four from each side.
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And on the right-hand side, we’d have 𝑥 is less than negative four.
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So we can say that if the value of 𝑥 is less than negative four, we come down the right-hand branch.
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However, if it’s greater than negative four we come down the left-hand branch.
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So if we’re coming down the left-hand branch, we know that the value of 𝑥 plus four is going to be positive.
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So we actually don’t need to worry, and what we can write is 𝑥 plus four equals 𝑥 plus four.
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So what we’re actually saying is that 𝑥 plus four on the left-hand side is equal to 𝑥 plus four on the right-hand side.
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So we could just think at this point, ah well, it’s true for all values of 𝑥 obviously.
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However, what we do need to remember is the range for the values of 𝑥 that we identified.
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And that is that 𝑥 has got to be greater than or equal to negative four.
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So therefore, we know to satisfy this equation, we can say that 𝑥 is gonna be all values of 𝑥 that are greater than or equal to negative four.
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So moving down the right-hand channel, we know that 𝑥 is less than negative four and we also know that 𝑥 plus four is giving us a negative value.
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So therefore, what we’re gonna have to do is take a negative of that negative to turn it into a positive.
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So therefore, what we’re gonna have to work with is negative 𝑥 plus four is equal to 𝑥 plus four.
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So we’re gonna get negative 𝑥 minus four equals 𝑥 plus four.
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So then if we add 𝑥 to both sides of the equation, we’re gonna get negative four equals two 𝑥 plus four.
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And then if we subtract four, we get negative eight equals two 𝑥.
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And then if I divide both sides by two, I get the answer 𝑥 is equal to negative four.
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Well, we might think that 𝑥 equals negative four wouldn’t strictly be speaking correct because if we look it down the right-hand side, we actually said that 𝑥 has got to be less than negative four.
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However, actually, if we had the negative of zero, it would still be positive, so it’d still be zero.
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And we actually cover this solution in the 𝑥 is greater than or equal to negative four anyway.
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So therefore, we could say that the solution is 𝑥 is greater than or equal to negative four.
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Or we can actually write this also in interval notation.
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And if we did that, we’d have left closed interval negative four, comma ∞, right open interval.
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And what this tells us is that the negative four can be part of our solution set, whereas the ∞ will not be.