Absolute value is a mathematical function that takes the positive version of whatever number is inside the absolute value signs, which are drawn as two vertical bars. For example, the absolute value of -2 -- written as |-2| -- is equal to 2. In contrast, linear equations describe the relationship between two variables. For example, y = 2x +1 tells you that to calculate y for any given value of x, you double the value of x and then add 1.
Domain and Range
Domain and range are mathematical terms which describe all of the possible input (x) values and all of the possible output (y) values, respectively, of a function. Any numbers can be input into an absolute value or linear equation, and so the domains of both include all real numbers. Because absolute values cannot be negative, their smallest possible value is zero. In contrast, linear equations can describe values that are negative, zero or positive. As a result, the range of an absolute value function is zero and all positive numbers, while the range of a linear equation is all numbers.
The graph of an absolute value function looks like a "v." The tip of the "v" is located at the minimum y-value of the function (unless there is a negative sign in front of the absolute value bars, in which case the graph is an upside-down "v" with the tip at the function's maximum y-value). In contrast, the graph of a linear equation is a straight line described by the equation y = mx + b, where m is the slope of the line and b is the y-intercept (i.e. where the line crosses the y axis).
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Number of Variables
Absolute value equations can contain two variables, just like linear equations do, but they can also contain just one variable. For example, y = |2x| + 1 is a graph of an absolute value equation similar to the linear equation y = 2x +1 in format (though the graphs look quite different, as described above). An example of an absolute value equation with only one variable is |x| = 5.
Linear equations and two-variable absolute value equations contain two variables and therefore cannot be solved without also having a second equation. For absolute value equations with one variable, there are usually two solutions. In the absolute value equation |x| = 5, the solutions are 5 and -5, since the absolute value of each of those numbers is 5. A more complicated example is as follows: |2x + 1| -3 = 4. To solve an equation like this, first rearrange it so that the absolute value is by itself on one side of the equal sign. In this case, that means adding 3 to both sides of the equation. This yields |2x + 1| = 7. The next step is to remove the absolute value bars and set one version equal to the original number, 7, and the other version equal to the negative value of that, i.e. -7. Lastly, solve each expression separately. So, in this example we have 2x + 1 = 7 and 2x + 1 = -7, which simplifies to x = 3 or -4.