Question

A doctor has found that, over the years 95% of the babies he has delivered weighed x pounds, where $|x - 8.1| \leqslant 1.6$what range of weights corresponds to this inequality

Answer 1

Inequalities and absolute value

We want to solve the following inequality for x:

$|x-8.1|\leq1.6$

First, let's analyze the possibilities for the absolute value |x - 8.1|. We know that the absolute value is referred always to a positive quantity, then

$0\leq|x-8.1|$

We have two possible cases here:

$\begin{gathered} (1).0\leq+(x-8.1) \\ (2).0\leq-(x-8.1) \end{gathered}$

We are going to analyze the first case

$0\leq x-8.1\leq1.6$

We add 8.1 on the three sides:

$\begin{gathered} +8.1\leq x\leq1.6+8.1 \\ 8.1\leq x\leq9.7 \end{gathered}$

Now, we are going to analyze the second case

$\begin{gathered} 0\leq-(x-8.1)\leq1.6 \\ 0\leq-x+8.1\leq1.6 \end{gathered}$

Substracting 8.1 on the three sides:

$\begin{gathered} \\ -8.1\leq-x\leq1.6-8.1 \\ -8.1\leq-x\leq-6.5 \end{gathered}$

Now we multiply by -1 on the three sides ( since it is a negative number multiplication the inequality signs change their direction):

$\begin{gathered} -1(-8.1)\ge-1(-x)\ge-1(-6.5) \\ 8.1\ge x\ge6.5 \end{gathered}$

Then, we got two answers from both cases:

$\begin{gathered} 6.5\leq x\leq8.1 \\ \& \\ 8.1\leq x\leq9.7 \end{gathered}$

Combining them we have a final answer:

Answer: 6.5 ≤ x ≤ 9.7. The range of weights is from 6.5 pounds to 9.7 pounds