Question

If $4x - 12 \geqslant x + 9$, which of the following must be true?
(A) $x > 6$
(B) $x < 7$
(C) $x > 7$
(D) $x > 8$
(E) $x < 8$

Answer 1

Hint: Here we solve this inequality by adding and subtracting the required terms on both sides. The variable term can be added or subtracted with the variable term only and the constant term can be added and subtracted with the constant term only.

Complete step by step answer:
In the above question, it is given that
$4x - 12 \geqslant x + 9$
Now, we will subtract x on both the sides
$ \Rightarrow 4x - x - 12 \geqslant x - x + 9$
$ \Rightarrow 3x - 12 \geqslant 9$
Now, we will add $12$on both the sides
$ \Rightarrow 3x - 12 + 12 \geqslant 9 + 12$
$ \Rightarrow 3x \geqslant 21$
Now, we will divide both side by $3$
$ \Rightarrow \dfrac{{3x}}{3} \geqslant \dfrac{{21}}{3}$
$ \Rightarrow x \geqslant 7$
This means that all the values which are greater than or equal to 7 will satisfy the inequality. So, any number we pick should satisfy this condition.
If we look at the given options, option (A): $x>6$ is always correct. Because any value we take must always be greater than or equal to 7 to satisfy the given inequality. So, that value will definitely be greater than 6.
Option (B): $x<7$ doesn’t satisfy the required condition.
Option (C): $x>7$ This partially satisfies the required condition. If we take $x=7$, this condition fails. So, this can’t always be true.
Option (D): $x>8$ This condition can’t always be true. If we take $x=7.2$ this condition fails.
Therefore, option (A) is correct.

Note:
It should be noted that when you perform multiplication or division by a negative, you have to reverse the inequality sign as well. Also, we cannot multiply or divide both sides by zero. It is an example of a linear equation in one variable. A linear equation in one variable holds only one variable and whose highest index of power is $1$.