Question

If the replacement set is the set of natural numbers. Solve:
(i) x – 5 < 0
(ii) x + 1 $\le $ 7
(iii) 3x – 4 > 6
(iv) 4x + 1 $\ge $ 17

Answer 1

Hint: To solve this question, we will solve each inequality. To solve an inequality, we will perform manipulations on the inequality to bring the variable on one side of the sign of inequality and all the other numbers at the other side of the sign of inequality. Once this is done, we can find the solution set for x.

Complete step by step answer:
The first inequality given to is x – 5 < 0.
We will add 5 to both sides of the inequality.
$\Rightarrow $ x – 5 + 5 < 0 + 5
$\Rightarrow $ x < 5
Thus, the value of x must be less than 5.
Therefore, the solution set of x for the inequality x – 5 < 0 is $S=\left\{ x:x\in \left( -\infty ,5 \right) \right\}$.
The second inequality given to is x + 1 $\le $ 7.
We will subtract 1 from both sides of the inequality.
$\Rightarrow $ x + 1 – 1 $\le $ 7 – 1
$\Rightarrow $ x $\le $ 6
Thus, the value of x must be less than or equal to 6.
Therefore, the solution set of x for the inequality x + 1 $\le $ 7 is $S=\left\{ x:x\in \left( -\infty ,6 \right] \right\}$.
The third inequality given to is 3x – 4 > 6.
We will add 4 to both sides of the inequality.
$\Rightarrow $ 3x – 4 + 4 > 6 + 4
$\Rightarrow $ 3x > 10
Now, we will divide both sides by 3.
$\begin{align}
  & \Rightarrow \dfrac{3x}{3}>\dfrac{10}{3} \\
 & \Rightarrow x>\dfrac{10}{3} \\
\end{align}$
Thus, the value of x must be greater than $\dfrac{10}{3}$.
Therefore, the solution set of x for the inequality 3x – 4 > 6 is $S=\left\{ x:x\in \left( \dfrac{10}{3},\infty \right) \right\}$.
The fourth and final inequality given to is 4x + 1 $\ge $ 17.
We will subtract 1 from both sides of the inequality.
$\Rightarrow $ 4x + 1 – 1 $\ge $ 17 – 1
$\Rightarrow $ 4x $\ge $ 16
Now, we will divide both sides by 4.
$\Rightarrow \dfrac{4x}{4}\ge \dfrac{16}{4}$
$\Rightarrow $ x $\ge $ 4
Thus, the value of x must be greater than or equal to 4.
Therefore, the solution set of x for the inequality 4x + 1 $\ge $ 17 is $S=\left\{ x:x\in \left[ 4,\infty \right) \right\}$.

Note: The replacement set is defined as the set, from which the values of the variable which are involved in the inequation, are chosen, is known as replacement set. In the solution set, the square brackets “[]” means that the number is included and rounded brackets “()” means that the number is not included in the set.