How To Solve Problems With Absolute Value

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.

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

Distances in the universe can be measured in all directions.

In this wiki, we intend to discuss this techniques along with strategies on when to use which.

is not a possible solution, but it does not mean it's not a possible solution for Case 1 because we're simply going piece by piece in this piecewise function--in the end we will take the union of all possible solutions.

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.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.As such, it is useful to consider distance as an absolute value function.In this section, we will investigate absolute value functions.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 best that manufacturers can do is to try to guarantee that the variations will stay within a specified range, often ±1%, ±5%, or ±10%. Use the absolute value function to express the range of possible values of the actual resistance. The absolute value of the difference between the actual and nominal resistance should not exceed the stated variability, so, with the resistance \(R\) in ohms, \[|R−680|\leq34\] Exercise \(\Page Index\) Students who score within 20 points of 80 will pass a test.Write this as a distance from 80 using absolute value notation.Recall that in its basic form \(f(x)=|x|\), the absolute value function, is one of our toolkit functions.The absolute value function is commonly thought of as providing the distance the number is from zero on a number line.

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