Question

How do you solve $ \dfrac{2}{3}\left( 12-x \right)>4 $ ?

Answer 1

Hint: In this question, we have to find the value of x. It is given that there is an inequation; therefore, we did not get the exact answer of x but will get some range where the values of x lie. Thus, we solve this problem using basic mathematical rules. We first multiply both sides of the inequation by 3, and then apply the distributive property $ a(b-c)=ab-ac $ on the left-hand side of the inequation. After that, we will subtract 24 on both sides of the inequation and make the necessary calculations. Then, we will divide 2 on both sides of the equation, and again multiply (-1) on both sides. In the last, we change the greater-than sign to less than sign, to get the required solution for the problem.

Complete step by step answer:
According to the problem, we have to find the value of x.
The inequation given to us is $ \dfrac{2}{3}\left( 12-x \right)>4 $ ---------- (1)
So, we will start solving this problem by multiplying both sides of the inequation by 3, we get
 $ \Rightarrow \dfrac{2}{3}\left( 12-x \right).(3)>4.(3) $
As we know, the same terms in the division will cancel out, we get
 $ \Rightarrow 2\left( 12-x \right)>12 $
Now, we will apply the distributive property $ a(b-c)=ab-ac $ on the left-hand side of the above inequation, we get
 $ \Rightarrow 24-2x>12 $
Now, subtract 24 on both sides of the above inequation, we get
 $ \Rightarrow 24-2x-24>12-24 $
As we know, the same terms with opposite signs cancel out each other, therefore we get
 $ \Rightarrow -2x>-12 $
Now, we will divide both sides of the inequation by 2, we get
 $ \Rightarrow -\dfrac{2}{2}x>-\dfrac{12}{2} $
On further simplification, we get
 $ \Rightarrow -x>-6 $
Now, we will multiply (-1) on both sides of the above equation, we get
 $ \Rightarrow -x.(-1)>-6.(-1) $
Therefore, we will change the > sign to < sign because we are removing the negative sign, hence we get
 $ \Rightarrow x<6 $
Therefore, for the inequation $ \dfrac{2}{3}\left( 12-x \right)>4 $ , the value of x is less than 6, which implies x can take values $ ....-3,-2...1,2,..5 $ which is the required solution to the problem.

Note:
 While solving this problem, do mention all the formulas you are using to avoid confusion and mathematical errors. One of the alternative methods to solve this problem is to apply the distributive property $ a(b-c)=ab-ac $ at the beginning of the solution and then use mathematical rules to solve the problem, which is the required solution to the problem. Also, do not forget to change the greater than sign at the end of your solution.