Question

How do you write each specification as an absolute value inequality: 6.3 (is less than equal to) h (is less than or equal to) 10.3?

Answer 1

In the above mentioned question we need to write a final inequality from the inequality that has been mentioned in the question. For this we are going to cancel out the average value of the upper and lower value i.e. 6.3 and 10.3 then we are going to use the modulus function to form a resultant inequality.

Complete step-by-step solution:
In the above question that has been stated we know the upper and lower limit of the inequality which is $6.3\le h\le 10.3$ we need to make this interval in such a way that it will have an absolute value which can also be termed as modulus value of the whole function. For converting the given inequality in the question we will first make use of the upper and the lower limit by finding the average of the two values which will be:

& =\dfrac{6.3+10.3}{2} \\\ & =8.3 \\\ \end{aligned}$$ Now we will use this average value that we just found in the inequality given in the question by subtracting the average value in each part of the equation and hence we will get: $$\begin{aligned} & \Rightarrow 6.3-8.3\le h-8.3\le 10.3-8.3 \\\ & \Rightarrow -2\le h-8.3\le 2 \\\ \end{aligned}$$ Now as the upper limit and the lower limit have the same numeric value we can convert it to its absolute value with the use of modulus. So the whole equation can also be seen as $$\begin{aligned} & h-8.3\le 2.........\left( 1 \right) \\\ & -\left( h-8.3 \right)\ge 2.......\left( 2 \right) \\\ \end{aligned}$$ So after merging the two equations we will get the third and final equation which is an absolute value which will be $$\left| h-8.3 \right|\le 2$$ **So the final absolute inequality of the equation will be $$\left| h-8.3 \right|\le 2$$** **Note:** In the above question first understand what the question is saying i.e. absolute value or inequality is the same as inequality with a modulus inequality which can also be termed as combining two inequalities to form one inequality as result.