Question

How do you rewrite the inequality $\left| {11 - 2x} \right| \geqslant 13$ as a compounded inequality?

Answer 1

Hint: In this question we are asked to rewrite the inequality with absolute as a compound inequality, first solve the inequality, 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 $\left| {11 - 2x} \right| \geqslant 13$,
Now as the absolute value is already isolated, so we will solve the positive version of the inequality, i.e.,
$ \Rightarrow 11 - 2x \geqslant 13$,
Subtract 11 to both sides of the inequality, we get,
$ \Rightarrow 11 - 2x - 11 \geqslant 13 - 11$,
Now simplify the equation we get,
$ \Rightarrow - 2x \geqslant 2$,
Now dividing both sides with 2, we get,
$ \Rightarrow \dfrac{{ - 2x}}{2} \geqslant \dfrac{2}{2}$,
Now simplifying we get,
$ \Rightarrow - x \geqslant 1$,
Now multiplying with negative sign, 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 - \left( { - x} \right) \leqslant - 1$,
Now simplifying we get,
$ \Rightarrow x \leqslant - 1$,
Now we will solve the negative version of the inequality, i.e.,
$ \Rightarrow - \left( {11 - 2x} \right) \geqslant 13$,
Now simplifying we get,
$ \Rightarrow - 11 + 2x \geqslant 13$,
Adding 11 to both sides of the inequality, we get,
$ \Rightarrow - 11 + 2x + 11 \geqslant 13 + 11$,
Now simplify the equation we get,
$ \Rightarrow 2x \geqslant 24$,
Now dividing both sides with 2, we get,
$ \Rightarrow \dfrac{{2x}}{2} \geqslant \dfrac{{24}}{2}$,
Now simplifying we get,
$ \Rightarrow x \geqslant 12$$ - 1$ ,
So, finally the solution can also be written as compounded as$x \leqslant - 1$ or $x \geqslant 12$.

Final Answer:
$\therefore $The given inequality $\left| {11 - 2x} \right| \geqslant 13$ is written in compounded inequality as $x \leqslant - 1$ or $x \geqslant 12$.

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.