Question

How do you solve and write the following in the interval notation: $3x-2<7$ and $-3x\le 15$?

Answer 1

Hint: From the given question we are asked to find the solution to the inequality and asked to find the interval. For solving this question we will first solve the two inequality using mathematical operations like division, multiplication and after doing that we will combine the both and get the interval notation. So, we proceed with our solution as follows.

Complete step by step solution:
For the first equation,
$\Rightarrow 3x-2<7$
we add two on both sides of the equation. So, we get,
$\Rightarrow 3x-2+2<7+2$
$\Rightarrow 3x<9$
Now we use the basic operation in mathematics that is division and divide both the sides of the equation with the integer three. So, we get the equation reduced as follows.
$\Rightarrow \dfrac{3x}{3}<\dfrac{9}{3}$
We cancel the three on the denominator and numerator on both sides of the equation. So, we get,
$\Rightarrow x<3$
For the second equation that is $-3x\le 15$,
$\Rightarrow -3x\le 15$
We will use the basic operation in mathematics that is division and divide both the sides of the equation with the integer negative of three. So, we get the equation reduced as follows.
$\Rightarrow \dfrac{-3x}{-3}\ge \dfrac{15}{-3}$
We cancel the negative of three on the denominator and numerator on both sides of the equation. So, we get,
$\Rightarrow x\ge -5$
So, the solution is $x \ge -5$ and $ x<3$
The interval notation is $[-5,3)$.

Note: Students must be very careful in doing the calculations. Students must have good knowledge in basic mathematical operations and also inequality and its properties. Students should not do mistake in inequality like for example if we write the next step as $ \dfrac{-3x}{-3}\le \dfrac{15}{-3}$ instead of $\dfrac{-3x}{-3}\ge \dfrac{15}{-3}$ then our solution will be wrong.