Question

How‌ ‌do‌ ‌you‌ ‌solve‌ ‌\[\dfrac{x+7}{x-4}<0\]?‌

Answer 1

Hint: We know that an inequality compares two values, showing if one is less than, greater than, or simply not equal to another value. By having a clear idea about inequality we can solve this problem. We need to involve all the conditions which are required to solve this problem.

Complete step-by-step solution:
For the given question we are given to solve the inequality equation \[\dfrac{x+7}{x-4}<0\]. So let us consider the equation as equation (1).
\[\dfrac{x+7}{x-4}<0....................\left( 1 \right)\]
By observing the equation (1) we can see that the numerator is less than 0 or negative.
If the value of x is -7 then it is equal to 0. So, therefore the value of x should be greater than -7. Let us prove it mathematically.
Let us call equation (1),
\[\dfrac{x+7}{x-4}<0\]
Now by multiplying with x-4 on both sides of equation (1), we get
\[\Rightarrow \dfrac{\left( x+7 \right)\left( x-4 \right)}{\left( x-4 \right)}<0.\left( x-4 \right)\]
\[\Rightarrow \left( x+7 \right)<0\]
By adding with -7 on both sides, we get
\[\Rightarrow x+7-7<-7\]
\[\Rightarrow x<-7\]
Therefore let us consider the above equation as equation (2).
\[x<-7.....................\left( 2 \right)\]
However, if the top and bottom of the equation are negative then it will become positive, and any x values less than -7 will give the positive values. However if x<4, then the bottom value will be negative, going to a negative value.
Therefore we can represent it as
\[\Rightarrow x-4>0\]
\[\Rightarrow x>4\]
Let us consider it as equation (3).
\[x>4................\left( 3 \right)\]
Therefore, from equation (2) and equation (3) we get
\[\Rightarrow 4 < x < -7\]
Let us consider it as equation (3), we get
\[4 < x < -7......................\left( 4 \right)\]
Therefore, equation (4) will be the solution for the given problem.

Note: Students should have a clear view on the difference between equality and inequality. By having a clear view, we can solve this problem. Students should not have any calculation mistakes while solving this problem. These mistakes should be avoided to solve this problem in a correct manner.