Question

List the solution set of $50 - 3(2x - 5) < 25$, given that $x \in W$. Also represent the solution set obtained on a number line.

Answer 1

Possible solutions of an inequality can be found by simplifying and rearranging the terms. Operations can be done on both sides of an inequality. Thus, we can find the range of $x$, which is the required solution set.

Formula used: Let $x$ and $y$ be any two numbers. Then,
$x < y \Rightarrow$ $x + a < y + a$ for any value $a$
also $x - a < y - a$ for any value $a$
but $ax < ay$ if $a > 0$ and $ax > ay$ if $a < 0$
similarly, $\dfrac{x}{a} < \dfrac{y}{a}$ if $a > 0$ and $\dfrac{x}{a} > \dfrac{y}{a}$ if $a < 0$

Complete step-by-step answer:
Given the inequality $50 - 3(2x - 5) < 25$
We need to find the solution set of the inequality.
That is, to find what all values $x$ can take under this condition.
Opening bracket on the left-hand side,
$50 + - 3 \times 2x + - 3 \times - 5 < 25$
$\Rightarrow 50 - 6x + 15 < 25$
Since $50 + 15 = 65$ we have,
$\Rightarrow 65 - 6x < 25$
We can add or subtract the same number on both sides of an inequality.
Subtracting $25$ from both the sides,
$65 - 6x - 25 < 25 - 25$
On simplification we get,
$40 - 6x < 0$
Subtracting $40$ from both sides we get,
$40 - 6x - 40 < \- 40$
Rearranging and simplifying,
$- 6x < \- 40$
Dividing by a negative number on both sides will reverse the inequality.
So, dividing by $- 1$ on both sides,
$\Rightarrow 6x > 40$
Dividing by $6$ on both sides,
$\Rightarrow x > \dfrac{{40}}{6}$
$\Rightarrow x > 6.666...$
In the question it is given that $x \in W$, means $x$ is a whole number.
Smallest whole number greater than $6.666...$ is $7$.
Therefore, the smallest possible value of $x$ is $7$.
Then any whole number greater than $7$ is a solution.