Question

How do you write the compound inequality as an absolute value inequality: $ 1.3 \leqslant h \leqslant 1.5? $

Answer 1

Hint: As we know that a compound inequality is an inequality that combines two simple inequalities as we can see that there are two inequality terms in the above question. The absolute value of a number is the distance of a number on the number line from $ 0 $ without taking into the consideration of direction from zero in the number line.

Complete step-by-step answer:
As per the given question we have a compound inequality $ 1.3 \leqslant h \leqslant 1.5 $ , here in this term the numbers $ 1.3 $ and $ 1.5 $ are called the extremes. of the inequality. To write the above compound inequality as an absolute value inequality we need to find the midpoint between the extremes of the inequality.
So we get: $ \dfrac{{1.3 + 1.5}}{2} $ , it gives $ \dfrac{{2.8}}{2} = 1.4 $ . Now we will subtract the value from each side of the equation until we reduce it to an absolute value inequality. Therefore we get, $ 1.3 - 1.4 \leqslant h - 1.4 \leqslant 1.5 - 1.4 $ .On further simplifying it gives $ - 0.1 \leqslant h - 1.4 \leqslant 0.1 $ .
We can write it as $ \left| {h - 1.4} \right| \leqslant 0.1 $ .
Hence the required absolute value identity is $ \left| {h - 1.4} \right| \leqslant 0.1 $ .
So, the correct answer is “ $ \left| {h - 1.4} \right| \leqslant 0.1 $ ”.

Note: We should note that the absolute value of a number is always positive excluding zero as zero is neither positive nor negative. While solving this type of question we should be careful in adding and subtracting the values as in inequality there are already positive and negative signs available and one should be careful because one wrong sign can give the wrong answers as the inequality will change. WE should perform each step carefully in order to avoid confusion and calculation mistakes.