Question

How do you solve compound inequalities $4x < 16$ or $12x > 144?$

Answer 1

As we know that a compound inequality is an inequality that combines two simple inequalities as we can see that there are two inequality terms in the above question. WE can solve compound inequalities by adding, subtracting, multiplying or dividing both sides whichever is suitable until we are left with the variable only, but these all tend to change the direction of inequality.

Complete step-by-step solution:
As per the given question we have inequalities $4x < 16$or $12x > 144$. We will take the first part, here we have $4x < 16$, by dividing both the sides by $4$we get: $\dfrac{{4x}}{4} < \dfrac{{16}}{4}$ $\Rightarrow x < 4$.

Now we take the second part and divide both the sides by $12$, we obtain $\dfrac{{12x}}{{12}} > \dfrac{{144}}{{12}} \Rightarrow x > 12$. So we now have two new inequalities. From both we get that $x < 4,x > 12$, it means that $x$ must be less than $4$ or $x$ must be greater than $12$. We can write it as $\\{ x\left| {x < 4,x > 12} \right|\\}$.

Hence the required answer is $\\{ x\left| {x < 4,x > 12} \right|\\}$.

Note: While solving this type of question we should be careful in adding and subtracting the values as in inequality there are already positive and negative signs available and one should be careful because one wrong sign can give the wrong answers as the inequality will change. We should perform each step carefully in order to avoid confusion and calculation mistakes.