Question

How do you solve absolute inequalities?

Answer 1

Hint: In this question, we have to find the general way of solving the absolute inequalities, absolute inequalities are the equations in which an absolute value function is given either smaller than or greater than some other quantity. A function that always gives a non-negative number as the answer is known as an absolute value function, that is, if the input value is a negative number, the absolute value function (also called modulus function) converts it into a positive function while a positive input value remains a positive value. $ \left| x \right| = x $ when x is positive and $ \left| x \right| = - x $ when x is negative; thus an absolute is a piecewise function that can be written as two subfunctions. We can solve the given question using the above information.

Complete step-by-step answer:
Let an absolute value function be $ \left| {f(x)} \right| $ that is smaller than a real number “a”
 $ \left| {f(x)} \right| < a $
Now we know that
 $ \left| {f(x)} \right| = f(x) $ or $ \left| {f(x)} \right| = - f(x) $
 $ \Rightarrow f(x) < a $ or $ - f(x) < a $
Applying the rules of inequality, we get –
 $ f(x) < a $ or $ f(x) > - a $
Combining the above two inequalities, we get –
 $ - a < f(x) < a $
Hence for solving the absolute inequalities, we take both the positive and negative possible values of the function.

Note: A function is expressed in terms of a variable quantity represented by an alphabet, the value of the function changes as the value of the variable changes. Input or index values are the values of the variable; in this question x is the variable quantity. From these equations, we can find the domain and range of the equation, and we can plot the equation on the graph too. For example, let absolute inequality be $ \left| {x + 7} \right| < 9 $ . it can be written as $ x + 7 < 9 $ or $ x + 7 > - 9 $
 $ \Rightarrow x < 9 - 7 $ or $ x > - 9 - 7 $
 $ \Rightarrow - 16 < x < 2 $
Thus, $ x \in ( - 16,2) $ .