Question

If \[r\] is a real number such that \[\left| r \right| < 1\] and if \[a = 5\left( {1 - r} \right)\], then what is the interval of \[a\]?
A. \[0 < a < 5\]
B. \[ - 5 < a < 5\]
C. \[0 < a < 10\]
D. \[0 \le a < 10\]
E. \[ - 10 < a < 10\]

Answer 1

Hint: Simplify the absolute value inequality function. Then substitute each value of \[r\] in the given equation. Find the minimum and maximum value of \[a\] to reach the required answer.

Formula used:
Absolute value inequality: If \[\left| a \right| < b\], then \[ - b < a < b\].

Complete step by step solution:
The given absolute value inequality function is \[\left| r \right| < 1\] and the equation is \[a = 5\left( {1 - r} \right)\].
Let’s simplify the inequality function \[\left| r \right| < 1\].
Apply the rule of absolute value inequality.
\[ - 1 < r < 1\]
Now substitute \[r = - 1\] in the given equation \[a = 5\left( {1 - r} \right)\].
\[a = 5\left( {1 - \left( { - 1} \right)} \right)\]
\[ \Rightarrow \]\[a = 5\left( 2 \right)\]
\[ \Rightarrow \]\[a = 10\]
Now substitute \[r = 1\] in the given equation \[a = 5\left( {1 - r} \right)\].
\[a = 5\left( {1 - 1} \right)\]
\[ \Rightarrow \]\[a = 5\left( 0 \right)\]
\[ \Rightarrow \]\[a = 0\]
Using the inequality equation, we get
\[0 < a < 10\]
Hence the correct option is C.

Note: The absolute value of a number is the distance between the number and the origin on a number line.
Absolute value inequality:
If \[\left| a \right| < b\], then \[ - b < a < b\]
If \[\left| a \right| > b\], then \[a < - b\] or \[a > b\]