Question

How do you solve $\left| {2x + 7} \right| > 23$?

Answer 1

Hint: In order to solve this question, we first need to isolate our absolute value term. After that we establish a relation between the positive and the negative extent of the constant and solve accordingly to get our required answer.

Complete step-by-step solution:
In this question, we are asked to solve an absolute inequality. An absolute function always uses a non-negative form.
To solve this question, we need to consider both the positive and negative extent of the given constant. Thus,
$ \Rightarrow - 23 > \left| {2x + 7} \right| > 23$
Let us divide this inequality in two parts and solve separately:
$ \Rightarrow - 23 > \left| {2x + 7} \right|$ and $\left| {2x + 7} \right| > 23$
On adding $ - 7$ to both sides, we get:
$ \Rightarrow - 23 - 7 > 2x$ and $2x > 23 - 7$
On simplifying it further, we get:
$ \Rightarrow - 30 > 2x$ and $2x > 16$
Therefore, $ - 15 > x$ and $x > 8$

In interval notation, we have:
$ \Rightarrow \left( { - \infty , - 15} \right)$ and $\left( {8,\infty } \right)$

Note: In mathematics, inequality is simply a relation which makes a comparison between two unequal numbers. A rational inequality is simply an inequality containing rational expressions and having zero on one side. After solving inequalities, we can simply express them on number lines. Inequalities are expressed using the greater than’ <’ or less than ’ > ‘signs
Absolute value of a number can simply be defined as the non-negative value of that number without regard to its sign. It is denoted by the symbol $\left| x \right|$ and is also known as the modulus of that number. It can also simply mean the distance of that particular number from zero. So for example, if we have the number $ + 6$, then its distance from zero will be $6$. Similarly if we have the number $ - 6$, then its distance from zero will also be $6$ regardless of the sign.