Question

How do you solve the inequality $x+5\ge 2x+1$ and $-4x<-8$ ?

Answer 1

Hint:Here in this question we have been asked to solve the given inequality expressions $x+5\ge 2x+1$ and $-4x<-8$ . Now we will initially simplify the inequality expression $-4x<-8$ and get a value for the variable $x$ . Then use that and simplify the other expression of inequality.

Complete step by step solution:
Now considering from the question we have been asked to solve the given inequality expressions $x+5\ge 2x+1$ and $-4x<-8$ .
Now we will initially simplify the inequality expression $-4x<-8$ and get a value for the variable $x$ . Then use that and simplify the other expression of inequality.
Now for simplifying $-4x<-8$ , we will divide the whole inequality expression with $-4$ . By doing that we will have $\dfrac{-4x}{-4}<\dfrac{-8}{-4}\Rightarrow x>2$ . Since the inequality sign changes when we divide it with a negative number.
Now we have $x>2$ as ……………(i)
Now we will simplify the other inequality expression $x+5\ge 2x+1$ . We will transfer all the terms containing the variable $x$ on the left hand side and the other terms on the right hand side of the expression. By doing that we will have $x-2x\ge 1-5$ .
Now by performing simple arithmetic basic subtraction between like terms we will have $-x\ge -4$ .
Now we will divide the whole expression by $-1$ . By doing this we will have $\dfrac{-x}{-1}\ge \dfrac{-4}{-1}\Rightarrow x\le 4$ . Since the inequality sign changes when we divide it with a negative number.
Now we have $x\le 4$ …………..(ii)
Since it is mentioned that to solve both the inequality expressions $x+5\ge 2x+1$ and $-4x<-8$ , we have to take the intersection of $x>2$ and $x\le 4$ .
Therefore we can conclude that the value of $x$ can be given as $\left( 2,\infty \right)\cap \left( -\infty ,4 \right]\Rightarrow \left( 2,4 \right]$ .

Note: During answering questions of this type we should be sure with our concepts that we are going to apply in the process. Someone may confuse and write the value of $x$ as $\left( 2,\infty \right)$ or $\left( -\infty ,4 \right]$ both are wrong. We should consider the combination of both and remember it.