Question

How do you solve \[4n - 5(n - 3) > 3(n + 1) - 20\]?

Answer 1

Hint: Multiply the term outside the bracket with terms inside the brackets on both sides of the equation. Solve each side of the equation by adding or subtracting terms with the same variable or constants. Solve the given inequality i.e. bring all variable values together on one side and all constant values together on the other side. Cancel possible terms and write the equation in simplest form. In the end inequality will give the value of ‘n’ being greater than a constant value.

Complete step by step answer:
We are given the inequality \[4n - 5(n - 3) > 3(n + 1) - 20\] … (1)
Since the inequality has only one variable i.e. n, we will calculate the value of n.
Multiply the term outside the bracket with terms inside the brackets on both sides of the equation (1)
\[ \Rightarrow 4n - (5 \times n) - ( - 5 \times 3) > (3 \times n) + (3 \times 1) - 20\]
Calculate each of the products on both sides of the equation
\[ \Rightarrow 4n - 5n + 15 > 3n + 3 - 20\]
Simplify the values on both sides of the equation by adding or subtracting terms with same variables
\[ \Rightarrow - n + 15 > 3n - 17\]
Shift all variable values i.e. values along with ‘n’ to one side of the inequality i.e. say left side of the inequality and similarly shift all constant values to right hand side of the inequality
\[ \Rightarrow - n - 3n > - 17 - 15\]
Calculate the sum or difference on both sides of the equation
\[ \Rightarrow - 4n > - 32\]
Cancel same factors from both sides of the equation i.e. 4
\[ \Rightarrow - n > - 8\]
Now we multiply both sides of the equation by -1
\[ \Rightarrow - n \times - 1 < - 8 \times - 1\]
Calculate the products on both sides of the inequality
\[ \Rightarrow n < 8\]
So, we get the value of n that it is always less than 8

Solution of the inequality \[4n - 5(n - 3) > 3(n + 1) - 20\] is \[n < 8\]

Note: Many students make the mistake of not changing the sign when multiplying negative values to both sides of the equation, keep in mind we change the sign of the inequality when we multiply minus sign or a negative term to both sides of the inequality. Also, keep in mind when shifting values from one side of the equation to another side of the equation, always change sign from positive to negative and vice-versa else the complete calculation becomes wrong, students can see this concept as of subtracting or adding the same term to both sides in order to eliminate that term.