Question

How do you solve $\left| {1 - 2x} \right| \geqslant 13$ ?

Answer 1

Hint: In the given equation we have to find the absolute value of the inequality. First we will solve the mod of the given inequality. We know that $\left| a \right| = \pm a$, so we will apply this formula to find the absolute value of the inequality. Inequalities are used to compare the two values, showing if one is greater than, less than or simply not equal to another. These values could be numerical or algebraic or a combination of both. The algebraic inequalities are also called literal inequalities. We use different types of symbols to compare the values. The symbol $ \leqslant $ represent less than or equal to, $ \geqslant $ represent greater than or equal to, $ \ne $ represent not equal to, > represent greater than, < represent less than.

Complete step by step solution:
Step: 1 the given inequality is,
$\left| {1 - 2x} \right| \geqslant 13$
Apply the formula $\left| a \right| = \pm a$ to solve the inequality.
$ \Rightarrow 1 - 2x \geqslant \pm 13$
Here we will consider the positive value of the inequality.
$ \Rightarrow 1 - 2x \geqslant + 13$
Step: 2 now subtract $ - 13$ to both sides of the equation.
$ \Rightarrow 1 - 2x - 1 \geqslant + 13 - 1$
Now simplify the given inequalities to its simplest form.
$ \Rightarrow - 2x \geqslant 12$
Step: 3 divide by two to both sides of the inequality to solve the given inequality.
$ \Rightarrow \dfrac{{ - 2x}}{2} \geqslant \dfrac{{12}}{2}$
Simplify the given inequality to its simplest form.
$ \Rightarrow - x \geqslant 6$
Step: 4 Now multiply both sides of the inequality by minus one and solve the given inequality to find the value of $x$.
$ \Rightarrow - x \times - 1 \geqslant 6 \times - 1$
We know that, when we multiply the given inequality by a negative number then the direction of the inequality will be changed.
$ \Rightarrow x \leqslant - 6$

Final Answer:
Therefore the solution of the inequality is $x \leqslant - 6$.

Note:
Reverse the direction of the sign of inequality when multiplying with a negative number to both sides of the inequality. Addition of a positive number does not change the direction of the inequality. When solving a multi step inequality always remember to change the direction of inequality sign while multiplying or dividing with a negative number.