Question

Describe all numbers x that are at a distance of 14 from the number −5. Express this using absolute value notation.

Answer 1

The distance (d) between two numbers a and b equals the absolute value of their difference:

$d=|a-b|$

If a number x is at a distance of 1/4 from the number -5, then:

$|x-(-5)|=\frac{1}{4}$

Solve for x. Remember that when an equation involves a variable inside an absolute value, two cases must be considered: If the expression inside the absolute value is positive or if it is negative.

Case 1: x-(-5) is positive.

Then:

$|x-(-5)|=x-(-5)$

Solve for x:

$\begin{gathered} |x-(-5)|=\frac{1}{4} \\ \Rightarrow x-(-5)=\frac{1}{4} \\ \Rightarrow x+5=\frac{1}{4} \\ \Rightarrow x=\frac{1}{4}-5 \\ \therefore x=-\frac{19}{4} \end{gathered}$

Case 2: x-(-5) is negative.

Then:

$|x-(-5)|=-(x-(-5))$

Solve for x:

$\begin{gathered} |x-(-5)|=\frac{1}{4} \\ \Rightarrow-(x-(-5))=\frac{1}{4} \\ \Rightarrow-(x+5)=\frac{1}{4} \\ \Rightarrow x+5=-\frac{1}{4} \\ \Rightarrow x=-\frac{1}{4}-5 \\ \therefore x=-\frac{21}{4} \end{gathered}$

Therefore, all the numbers that are at a distance of 1/4 from the number -5 are -21/4 and 19/4. They can be described by the equation:

$|x-(-5)|=\frac{1}{4}$