How To Solve Problems With Absolute Value

The distance from \(x\) to 5 can be represented using the absolute value as \(|x−5|\).We want the values of \(x\) that satisfy the condition \(| x−5 |\leq4\).

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.

Today, astronomers can detect galaxies that are billions of light years away.

Distances in the universe can be measured in all directions.

Analysis Note that \[\begin -4&x-5 & x-5&\leq4 \\1&x & x&9 \end\] So \(|x−5|\leq4\) is equivalent to \(1x\leq9\).

However, mathematicians generally prefer absolute value notation.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.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.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.To get each piece, you must figure out the domain of each piece.This method is highly beneficial when the question writer asks for the number of solutions instead of the actual solutions.Algebraically, for whatever the input value is, the output is the value without regard to sign.Example \(\Page Index\): Determine a Number within a Prescribed Distance Describe all values \(x\) within or including a distance of 4 from the number 5.