Question

How do you solve and write the following in interval notation: $-3\le \dfrac{x+4}{3}\le -1?$?

Answer 1

To solve the above question we will use the concept of algebraic inequality. To find the range of ‘x’ we will first cross multiply the given inequality by 3 and then we will subtract each term of the inequality with 4 so that 4 get eliminated from the middle.

Complete step-by-step solution:
We will use the concept of algebraic inequality to solve the above question. At first, we will eliminate fraction from the inequality by multiplying all terms with 3 then we will get:
$\Rightarrow -3\times 3\le \dfrac{x+4}{3}\times 3\le -1\times 3$
$\Rightarrow -9\le x+4\le -3$
Now, we will subtract 4 with all the terms of the given inequality. Then, we will get:
$\Rightarrow -9-4\le x+4-4\le -3-4$
Now, after simplifying the constant we will get:
$\Rightarrow -13\le x\le -7$
Since, we know that value x belongs from -13 to -7 and also includes -13 and -7.
So, we can write in the interval as: $\left[ -13,-7 \right]$ , square bracket means that the end terms are included in the interval.
Hence, the interval notation for $-3\le \dfrac{x+4}{3}\le -1$ is $\left[ -13,-7 \right]$.
This is our required solution.

Note: Students are required to note that when we multiply or divide an inequality by any positive number, there is no change in direction of inequality but when we multiply or divide an inequality by a negative number then there is change in direction of the inequality, so we should be careful when we multiply by negative because there is always a source of error.