Question

How do you solve the compound inequality $ 4x - 6 > 14 $ for $ 2x - 1 < - 5? $

Answer 1

Hint: Here we will solve the compound inequality one by one using the basic mathematical concepts and making the unknown term “x” subject and finding the resultant value accordingly.

Complete step-by-step solution:
Take the given expression: $ 4x - 6 > 14 $
Move the constant term on the left hand side of the equation from the right hand side of the equation. When you move any term from one side to another then the sign of the term also changes. Positive term becomes negative and negative term becomes positive.
 $ \Rightarrow 4x > 14 + 6 $
Simplify the above expression.
 $ \Rightarrow 4x > 20 $
Term multiplicative on one side if moved to the opposite side then it goes to the denominator.
 $ \Rightarrow x > \dfrac{{20}}{4} $
Find the factors on the numerator of the above equation
 $ \Rightarrow x > \dfrac{{4 \times 5}}{4} $
Common factors from the numerator and the denominator cancel each other. Therefore remove from the numerator and the denominator.
 $ \Rightarrow x > 4 $ …. (A)
Now, take second inequality given $ 2x - 1 < - 5 $
Move the constant term on the left hand side of the equation from the right hand side of the equation. When you move any term from one side to another then the sign of the term also changes. Positive term becomes negative and negative term becomes positive.
 $ \Rightarrow 2x < - 5 + 1 $
Simplify among the like terms
 $ \Rightarrow 2x < - 4 $
Term multiplicative on one side if moved to the opposite side then it goes to the denominator.
 $ \Rightarrow x < - \dfrac{4}{2} $
Find the factors on the numerator of the above equation
 $ \Rightarrow x < - \dfrac{{2 \times 2}}{2} $
Common factors from the numerator and the denominator cancel each other. Therefore remove from the numerator and the denominator.
 $ \Rightarrow x < - 2 $ …. (B)
Hence, the required solution is $ x > 4 $ or $ x < - 2 $

Note: Always remember that when you move any term from one side to another then the sign of the term also changes. Be careful about the sign while doing simplification remember the basic rules- Addition of two positive terms gives the positive term. Addition of one negative and positive term, you have to do subtraction and give signs of bigger numbers, whether positive or negative. Addition of two negative numbers gives a negative number but in actual you have to add both the numbers and give a negative sign to the resultant answer.