Question

In the given figure, the number line represents the solution of inequality _____?

A) $2x - 4 < 16$
B) $2x - 6 < 10$
C) $2x - 6 > 12$
D) $2x - 4 > 16$

Answer 1

In the given question, the number line represents the solution set of an inequality and four linear inequalities are given in the options, from which one inequality has the solution set which is represented in the given number line. We have to find which inequality has the solution set that is represented in the given number line. We will solve all the inequalities given in the question one by one and cross check with the given graph to get the required solution.

Complete step by step answer:
The given number line is

The above number line represents all the values which are less than 8 (not including 8). This means the solution set is $\left( { - \infty ,8} \right)$. We will check every option and find the correct match.
A) $2x - 4 < 16$
Add $4$ to both the sides
$\Rightarrow 2x - 4 + 4 < 16 + 4$
$\Rightarrow 2x < 20$
On division, we get
$\Rightarrow x < 10$
i.e., all the real numbers that are less than $10$, are the solutions of the given inequality. Hence, the solution set is $\left( { - \infty ,10} \right)$.

B) $2x - 6 < 10$
Add $6$ to both the sides
$\Rightarrow 2x - 6 + 6 < 10 + 6$
$\Rightarrow 2x < 16$
On division, we get
$\Rightarrow x < 8$
i.e., all the real numbers that are less than $8$, are the solutions of the given inequality. Hence, the solution set is $\left( { - \infty ,8} \right)$.
As we can see the given number line represents the solution set of inequality whose solution set is $\left( { - \infty ,8} \right)$, so we can conclude that this inequality is a correct answer.

C) $2x - 6 > 12$
Add $6$ to both the sides
$\Rightarrow 2x - 6 + 6 > 12 + 6$
$\Rightarrow 2x > 18$
On division, we get
$\Rightarrow x > 9$
i.e., all the real numbers that are greater than $9$, are the solutions of the given inequality. Hence, the solution set is $\left( {9,\infty } \right)$.

D) $2x - 4 > 16$
Add $4$ to both the sides
$\Rightarrow 2x - 4 + 4 > 16 + 4$
$\Rightarrow 2x > 20$
On division, we get
$\Rightarrow x > 10$
i.e., all the real numbers that are greater than $10$, are the solutions of the given inequality. Hence, the solution set is $\left( {10,\infty } \right)$.

Therefore, option (B) is the only correct option.

Note:

Remember that if the inequality involves $>$ or $<$, we draw an open circle (O) on the number line to indicate that the number corresponding to the open circle is excluded from the solution set and if the inequality involves $\geqslant$ or $\leqslant$, we draw a filled circle ($\cdot$) on the number line to indicate that the number corresponding to the filled circle is included in the solution set.
Also remember that addition or subtraction of the same number to both sides of an inequation doesn’t affect the sign of inequality.
Both sides of an inequation can be multiplied or divided by the same positive real number without changing the sign of inequality but the sign of inequality is reversed when both sides of an inequation are multiplied or divided by a negative number.
Any term of an inequality can be taken to the other side with its sign changed without affecting the sign of inequality.