Question

A hot dog factory must ensure that its hot dogs are between $6\dfrac{1}{4}$ inches and $6\dfrac{3}{4}$inches in length. If $h$is the length of a hot dog from this factory, then which of the following inequalities correctly, expresses the accepted values of $h$?
A) \[\left| {h - 6\dfrac{1}{2}} \right| < \dfrac{1}{2}\]
B) \[\left| {h - 6\dfrac{1}{4}} \right| < \dfrac{1}{2}\]
C) \[\left| {h - 6\dfrac{1}{2}} \right| < \dfrac{1}{4}\]
D) \[\left| {h - 6\dfrac{1}{4}} \right| < \dfrac{1}{4}\]

Answer 1

Hint: First write the given condition in the mathematical form which will be in the form of inequality.
Use the property of the modulus function which states that if $|x| < a$ then $ - a < x < a$

Complete step-by-step answer:
We are given that the length $h$ of the hot dogs are between $6\dfrac{1}{4}$ inches and $6\dfrac{3}{4}$inches.
We have to express it in the form of inequality.
First, we write the given condition in the mathematical form.
That is, $6\dfrac{1}{4} < h < 6\dfrac{3}{4}$
Simplify the inequality.
$\dfrac{{25}}{4} < h < \dfrac{{27}}{4}.......(1)$
Now we check each option, the option which gives the condition (1) will be our answer.
Option -A
Given that \[\left| {h - 6\dfrac{1}{2}} \right| < \dfrac{1}{2}\]
Use the property of the modulus function which states that if $|x| < a$ then $ - a < x < a$
Our inequality will be,
$\dfrac{{ - 1}}{2} < h - 6\dfrac{1}{2} < \dfrac{1}{2}$
Solve the inequality.
$
  \dfrac{{ - 1}}{2} < h - \dfrac{{13}}{2} < \dfrac{1}{2} \\
  \dfrac{{ - 1}}{2} + \dfrac{{13}}{2} < h - \dfrac{{13}}{2} + \dfrac{{13}}{2} < \dfrac{1}{2} + \dfrac{{13}}{2} \\
  6 < h < 7 \\
 $
This does not give the inequality (1) therefore, this cannot be the answer.
Option -B
Given that \[\left| {h - 6\dfrac{1}{4}} \right| < \dfrac{1}{2}\]
Use the property of the modulus function which states that if $|x| < a$ then $ - a < x < a$
Our inequality will be,
$\dfrac{{ - 1}}{2} < h - 6\dfrac{1}{4} < \dfrac{1}{2}$
Solve the inequality.
$
  \dfrac{{ - 1}}{2} < h - \dfrac{{25}}{4} < \dfrac{1}{2} \\
  \dfrac{{ - 1}}{2} + \dfrac{{25}}{4} < h - \dfrac{{25}}{4} + \dfrac{{25}}{4} < \dfrac{1}{2} + \dfrac{{25}}{4} \\
  \dfrac{{23}}{4} < h < \dfrac{{25}}{4} \\
 $
This does not give the inequality (1) therefore, this cannot be the answer.
Option- C
Given that \[\left| {h - 6\dfrac{1}{2}} \right| < \dfrac{1}{4}\]
Use the property of the modulus function which states that if $|x| < a$ then $ - a < x < a$
Our inequality will be,
$\dfrac{{ - 1}}{4} < h - 6\dfrac{1}{2} < \dfrac{1}{4}$
Solve the inequality.
$
  \dfrac{{ - 1}}{4} < h - \dfrac{{13}}{2} < \dfrac{1}{4} \\
  \dfrac{{ - 1}}{4} + \dfrac{{13}}{2} < h - \dfrac{{13}}{2} + \dfrac{{13}}{2} < \dfrac{1}{4} + \dfrac{{13}}{2} \\
  \dfrac{{25}}{4} < h < \dfrac{{27}}{4} \\
 $
This gives the inequality (1) therefore, this will be our answer.
Option -D
Given that \[\left| {h - 6\dfrac{1}{4}} \right| < \dfrac{1}{4}\]
Use the property of the modulus function which states that if $|x| < a$ then $ - a < x < a$
Our inequality will be,
$\dfrac{{ - 1}}{4} < h - 6\dfrac{1}{4} < \dfrac{1}{4}$
Solve the inequality.
$
  \dfrac{{ - 1}}{4} < h - \dfrac{{25}}{4} < \dfrac{1}{4} \\
  \dfrac{{ - 1}}{4} + \dfrac{{25}}{4} < h - \dfrac{{25}}{4} + \dfrac{{25}}{4} < \dfrac{1}{4} + \dfrac{{25}}{4} \\
  6 < h < \dfrac{{26}}{4} \\
 $
This does not give the inequality (1) therefore, this cannot be the answer.
Hence, option (C) is correct.

Note: We can solve this question by another method which is shown below.
The given inequality is
$6\dfrac{1}{4} < h < 6\dfrac{3}{4}$
Subtract $6\dfrac{1}{2}$from all the three parts.
$
  6\dfrac{1}{4} - 6\dfrac{1}{2} < h - 6\dfrac{1}{2} < 6\dfrac{3}{4} - 6\dfrac{1}{2} \\
  \dfrac{{25}}{4} - \dfrac{{13}}{2} < h - 6\dfrac{1}{2} < \dfrac{{27}}{4} - \dfrac{{13}}{2} \\
  \dfrac{{ - 1}}{4} < h - 6\dfrac{1}{2} < \dfrac{1}{4} \\
 $
Use the property of the modulus function which states that if $ - a < x < a$ then $|x| < a$.
Hence, \[\left| {h - 6\dfrac{1}{2}} \right| < \dfrac{1}{4}\]