Absolute value how many solutions

The absolute value of any number is positive. Here, we have the absolute value of something is negative. This is not possible so there are no possible x-values that make this equation true. Therefore, you can write:. The absolute value of any number is always positive. Use this to determine when there are no solutions to an absolute value equation. Notice that in both examples, the steps were the same as before.

You will always follow those two steps when solving any absolute value equation. You can think of the absolute value of any number as representing how far it is from zero on the number line. This is why the absolute value is always positive — it is representing a distance. This gives two possibilities:. So this is why we end up with two different equations. In the case of only one solution, you end up with an absolute value expression equal to zero. Since this means that the distance from zero on the number line is zero, you end up with only one equation.

When you study the graphs of absolute value equations, you can see the three cases of one solution, no solution, and two solutions graphically. Let's try to solve for not x first. We're just going to solve for the absolute value of x plus 7. You'll see what I mean. So I want to get all of the absolute values of x plus 7 on the left-hand side, so I want to get rid of this one on the right-hand side.

Easiest way to get rid of it is to add 6 times the absolute value of x plus 7 to the right-hand side. We can't, of course, only do that to the right-hand side. If these two things are equal and we are being told that they are, then if you add something on this side, the only way that the equality will hold is if you still do it on the left-hand side.

So let's do that, so plus 6 times the absolute value of x plus 7. And I want to get all of these constant terms on to the right-hand side. So I want to get rid of this positive 4. Easiest way is to subtract 4 right over there, but if we do it on the left-hand side, we have to do it on the right-hand side as well.

And so what does this get us? So our left-hand side, if I have 8 of something-- and in this case the something is absolute values of x plus 7's-- but if I have 8 of something and I add 6 of that same something, I now have 14 of that something.

So that's going to be 14 absolute values of x plus 7, 14 times the absolute value of x plus 7. The 4 and the negative 4 cancel out, and that was intentional. All courses. Algebra 1 Discovering expressions, equations and functions Overview Expressions and variables Operations in the right order Composing expressions Composing equations and inequalities Representing functions as rules and graphs. Algebra 1 Exploring real numbers Overview Integers and rational numbers Calculating with real numbers The Distributive property Square roots.

Algebra 1 How to solve linear equations Overview Properties of equalities Fundamentals in solving equations in one or more steps Ratios and proportions and how to solve them Similar figures Calculating with percents. Algebra 1 Visualizing linear functions Overview The coordinate plane Linear equations in the coordinate plane The slope of a linear function The slope-intercept form of a linear equation. Algebra 1 Formulating linear equations Overview Writing linear equations using the slope-intercept form Writing linear equations using the point-slope form and the standard form Parallel and perpendicular lines Scatter plots and linear models.