How To Solve Problems With Absolute Value

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Solution Using the variable \(p\) for passing, \(| p−80 |\leq20\) Figure 1.6.4 shows the graph of \(y=2|x–3| 4\).

The graph of \(y=|x|\) has been shifted right 3 units, vertically stretched by a factor of 2, and shifted up 4 units.

Example \(\Page Index\): Resistance of a Resistor Electrical parts, such as resistors and capacitors, come with specified values of their operating parameters: resistance, capacitance, etc.

However, due to imprecision in manufacturing, the actual values of these parameters vary somewhat from piece to piece, even when they are supposed to be the same.

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.

Absolute value equations are equations involving expressions with the absolute value functions.

This wiki intends to demonstrate and discuss problem solving techniques that let us solve such equations.

Solution We want the distance between \(x\) and 5 to be less than or equal to 4.

We can draw a number line, such as the one in , to represent the condition to be satisfied.


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