Question

How do you solve the absolute value inequality $ \left| {2x - 3} \right| < 5? $

Answer 1

Hint: Here we are given less absolute inequality pattern and so first of all we will clear the absolute value and accordingly follow the less than pattern. Will simplify the equations using the basic concepts and will find the value for the unknown term “x”.

Complete step-by-step solution:
Take the given expression: $ \left| {2x - 3} \right| < 5 $
Remove absolute mode which gives plus or minus values for the given expression as –
 $ - 5 < 2x - 3 < 5 $
Now, to get the equivalent expression we have to perform add or subtract terms to all the terms and focus here to get the value of “x” which is independent of other terms. Add to all the terms in the above expression.
 $ - 5 + 3 < 2x - 3 + 3 < 5 + 3 $
Simplify the above expression using the concepts that addition of two positive terms gives the positive term, Addition of one negative and positive term, you have to do subtraction and give sign of bigger number whether positive or negative and like terms with the same value and opposite sign cancel each other.
 $ - 2 < 2x < 8 $
Divide all the terms in the above expression by
 $ - \dfrac{2}{2} < \dfrac{{2x}}{2} < \dfrac{8}{2} $
Common factors from the numerator and the denominator cancels each other.
 $ - 1 < x < 4 $
This is the required solution.

Note: Be careful about the sign while doing simplification remember the golden rules-
i) Addition of two positive terms gives the positive term
ii) Addition of one negative and positive term, you have to do subtraction and give sign of bigger number whether positive or negative.
iii) Addition of two negative numbers gives a negative number but in actual you have to add both the numbers and give a negative sign to the resultant answer.