*Instead, the width is equal to 1 times the vertical distance as shown in Figure \(\Page Index\).*From this information we can write the equation \[\begin f(x)&=2|x-3|-2, \;\;\;\;\;\; \text \ f(x)&=|2(x-3)|-2, \;\;\; \text \end\] Analysis Note that these equations are algebraically equivalent—the stretch for an absolute value function can be written interchangeably as a vertical or horizontal stretch or compression.The distance from \(x\) to 5 can be represented using the absolute value as \(|x−5|\).

Let's work through some examples to see how this is done. Another benefit of this graphing technique is that you do not need to verify any of the solutions--since we are only graphing the pieces that are actually mathematically possible, we get all the solutions we are looking for, no less and no more.

If you could not discern the solutions from the picture, you can simply solve the equation for each case.

In this case, we can see that the isolated absolute value is to be less than or equal to a negative number.

Again, the absolute value will always be positive; hence, we can conclude that there is no solution.

Thus, the solutions are Sometimes absolute value equations have a ridiculous number of cases and it would take too long to go through every single case.

Therefore, we can instead graph the absolute value equations using the definition of absolute value as a piecewise function.

This means that the corner point is located at \((3,4)\) for this transformed function.

Solution The basic absolute value function changes direction at the origin, so this graph has been shifted to the right 3 units and down 2 units from the basic toolkit function. We also notice that the graph appears vertically stretched, because the width of the final graph on a horizontal line is not equal to 2 times the vertical distance from the corner to this line, as it would be for an unstretched absolute value function.

A very basic example would be as follows: if required.

However, these problems are often simplified with a more sophisticated approach like being able to eliminate some of the cases, or graphing the functions.

## Comments How To Solve Problems With Absolute Value

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