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A cyclist rides at an average speed of 31 miles per hour. The cyclist’s speed, s, can differ from the average by as much as 8 miles per hour. The absolute value inequality |31-s|<8 represents this situation. If the compound inequality -x<31-s and x>31-s also represents this situation, what is the value of x in the compound inequality?

Answer :

An absolute value inequality can have the following expression:

[tex]|P|It means that the absolute value of P is less than n.

For example, this expression is valid:

[tex]|2|<4[/tex]

But this expression is not

[tex]|7|<6[/tex]

We can write it mathematically as follows:

[tex]\begin{gathered} |P|-n \end{gathered}[/tex]

We are given the expression

[tex]|31-s|<8[/tex]

Which is equivalent to

[tex]\begin{gathered} 31-s<8\text{ and} \\ 31-s>-8 \end{gathered}[/tex]

Flipping the second inequality we have

[tex]-8<31-s[/tex]

Note when we compare with the given expression

[tex]-x<31-s[/tex]

We get the value of x = 8

The same happens when we flip the first one. We get the very same answer

x = 8

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