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Absolute Value Equations - Problem 3
Some absolute value equations may start off equaling a negative value, but before concluding that the equation has no solution, first isolate the absolute value equation to see how many, if any, solutions there will be. Remember that if the absolute value quantity is equal to a negative number, then there is no solution to the equation since an absolute value cannot equal a negative number. The first thing to do when solving an absolute value equation is to isolate the absolute value quantity using the same rules that apply to isolating the variable in an equation. In other words, eliminate everything else on the same side of the equal sign except for the absolute value quantity using inverse operations. Next, split the equation into two equations and remove the absolute value signs. One equation should be equal to the positive quantity on the other side, and the second equation should be equal to the negative quantity. However, if the quantity equals 0, then there will only be one equation to solve. Don't forget to check your solutions by substituting them back into the original equation.
A lot of students when they see a problem like this they think oh-oh! No solution because an absolute value can't be equal to a negative number. You can't have a negative distance. However, the absolute value has to be isolated before you look to see if there's going to be zero solutions or one solution or two solutions.
Like in this case I can't tell anything about how many answers I'll have until I get the absolute value by itself. So I'm going to have to add 4 to both sides and I'll have absolute value of 2x plus 3 equals to 0. Now since it's equal to 0 I know I'm only going to have one solution so it gets a little bit easier like when I'm separating this absolute value, I can't have something equal to 0 and also equal to -0 like what's -0? 2x minus 3 equals -0? Whatever, you don't even have to do that problem. My problem is just the extension here 2x plus 3 equals 0. Subtract 3 from both sides so you'll have 2x equals -3, divide both sides by 2 and you get x equals -3/2.
Again keep in mind x can be a negative number because I'm going to absolute value-ize it after I multiply by 2 and add 3.