Question

How do you solve the inequality $\left| {2x + 9} \right| < 25$?

Answer 1

Hint: In this question we are asked to solve the inequality with absolute, and these type of questions can be solved firstly by isolating the absolute expression to one side, as it is an absolute inequality it will have two versions i.e., positive and negative and then simplify the equation till we get the required result.

Complete step by step solution:
Inequalities are mathematical expressions involving the symbols >, <,$ \geqslant $,$ \leqslant $ To solve an inequality means to find a range, or ranges, of values that an unknown x can take and still satisfy the inequality.
The ​steps to solving an absolute value inequality​ are much like the steps to solving an absolute value equation:
​Step 1:​ Isolate the absolute value expression on one side of the inequality.
​Step 2:​ Solve the positive "version" of the inequality.
​Step 3:​ Solve the negative "version" of the inequality by multiplying the quantity on the other side of the inequality by −1 and flipping the inequality sign.
Now the give inequality is $\left| {2x + 9} \right| < 25$,
Now as the absolute value is already isolated, so we will solve the positive version of the inequality, i.e.,
$ \Rightarrow 2x + 9 < 25$,
Subtract 9 to both sides of the inequality, we get,
$ \Rightarrow 2x + 9 - 9 < 25 - 9$,
Now simplify the equation we get,
$ \Rightarrow 2x < 16$,
Now multiplying both sides with 2, we get,
$ \Rightarrow \dfrac{{2x}}{2} < \dfrac{{16}}{2}$,
Now simplifying we get,
$ \Rightarrow x < 8$,
Now we will solve the negative version of the inequality, i.e.,
$ \Rightarrow - \left( {2x + 9} \right) < 25$,
And when we apply negative sign for a greater than sign then it becomes less than and when we apply negative sign for a less than sign then it becomes greater than, then the equation becomes,
$ \Rightarrow 2x + 9 > - 25$,
Subtract 9 to both sides of the inequality, we get,
$ \Rightarrow 2x + 9 - 9 > - 25 - 9$,
Now simplify the equation we get,
$ \Rightarrow 2x > - 34$,
Now multiplying both sides with 2, we get,
$ \Rightarrow \dfrac{{2x}}{2} > \dfrac{{ - 34}}{2}$,
Now simplifying we get,
$ \Rightarrow x > - 17$,
So, finally the solution can also be written as $ - 17 < x < 8$.


The solution of the given inequality $\left| {2x + 9} \right| < 25$ is $ - 17 < x < 8$.

Note:
Because this problem involves an inequality with an absolute value function we must set up a system of inequalities because the absolute value function will transform a negative or positive number to a positive number.