Question

How do you solve compound inequalities $3x\ge 3$ or $9x < 54$ ?

Answer 1

Hint: To solve such types of problems, we first solve the individual inequality, which in this question are $3x\ge 3$ and $9x < 54$ . After that, we see whether the compound inequality contains “or” or “and”. Here, it has “or”, so we take both the solutions.

Complete step by step solution:
The definition of compound inequality is that it is a sentence with two inequality statements joined either by the conjunctions “or” or “and”. Compound inequalities are basically some simultaneous inequalities of the same variable. There may be various combinations of the inequalities. By combinations, we mean that the resultant solution can be the combined effect of all the equalities at the same time or the effect of one at a time. It is quite similar to the operations of sets. There can be a union of sets, or an intersection of them.
The compound inequality that we are given in this problem is,
$3x\ge 3$ or $9x < 54$
Let us first solve them one at a time. Starting off with $3x\ge 3$ , dividing both sides of the inequality by $3$ , we get,
$\Rightarrow x\ge 1$
Moving on to the other inequality $9x < 54$ , dividing both sides of the inequality by $9$ , we get,
$\Rightarrow x<6$
Now, the compound inequality contains the word “or”. Two mathematical expressions joined by the term “or” infers to a common solution which satisfies one of the two or both of the inequalities at a time. So, the overall solution will be the entire set of real numbers, or that $x\in \left( -\infty ,\infty \right)$ .

Therefore, we can conclude that the solution of the compound inequality will be $x\in \left( -\infty ,\infty \right)$ .

Note: The most important thing that we must watch out for here is the use of “or” or “and” and then proceed accordingly. The individual solutions and the final solution can also be shown on the number line. The blue line shows the solution $x\ge 1$ and the red one shows the solution $x < 6$ .