Question

How do you solve \[7\left( x+4 \right)>0\]?

Answer 1

Hint: This type of problem is based on the concept of inequality. First, we have to consider the whole function and then use distributive property on both the sides of the given equation, that is, \[a\left( b+c \right)=ab+ac\]. Then, we need to divide the whole obtained equation by 7. And then, subtract \[4\] from both the sides of the equation. Solve x considering the inequality.

Complete step by step answer:
According to the question, we are asked to solve the given equation \[7\left( x+4 \right)>0\].
We have been given the equation is \[7\left( x+4 \right)>0\] . -----(1)
We first have to use distributive property, that is, \[a\left( b+c \right)=ab+ac\] in equation (1).
We get,
 \[7x+4\left( 7 \right)>0\]
On further simplifications, we get,
\[7x+28>0\] --------(2)
Now, we have to divide the whole equation (2) by 7.
Therefore, \[\dfrac{7x+28}{7}>\dfrac{0}{7}\].
We know that,0 divided by any number is 0.
\[\Rightarrow \dfrac{7x+28}{7}>0\]
Let us now use the property \[\dfrac{a+b}{c}=\dfrac{a}{c}+\dfrac{b}{c}\] in the above equation.
We get, \[\dfrac{7x}{7}+\dfrac{28}{7}>0\]
\[\Rightarrow \dfrac{7x}{7}+\dfrac{7\times 4}{7}>0\]
Now, we have to look for the common terms.
Here 7 is the common term. Cancelling out 7 from the obtained equation, we get
\[x+4>0\]
We have to subtract 4 from both the sides of the obtained equation.
Therefore, we get \[x+4-4>0-4\].
We know that terms with the same magnitude and opposite signs cancel out.
We get, \[x>0-4\].
We also know that -4 subtracted from 0 is -4.
Using this to find the value of x, we get,
\[x>-4\]

Therefore, the value of x in the given equation \[7\left( x+4 \right)>0\] is \[x>-4\].

Note: Whenever you get this type of problem, we should always try to make the necessary changes in the given equation to get the final solution of the equation which will be the required answer. We should avoid calculation mistakes based on sign conventions. We should always make some necessary calculations to obtain zero in the right-hand side of the equation. We can also solve this problem by directly dividing the given equation by 7.
Therefore, we get \[\dfrac{7\left( x+4 \right)}{7}>\dfrac{0}{7}\].
On cancelling the common term 7 from the left side of the equation, we get \[x+4>0\]. Then, follow the above mentioned steps to obtain the final answer. This method is used to reduce the number of steps.