Question

Identify the set of solution such that in which interval the value of x exists, \[ - (x - 3) + 4 < 5 - 2x\]
A) \[( - \infty ,0)\]
B) \[( - \infty , - 1)\]
C) \[( - \infty , - 2)\]
D) \[( - \infty , - 5)\]

Answer 1

Hint: In this problem we have to find the interval value of an \[x\], by solving that \[ - (x - 3) + 4 < 5 - 2x\] given inequality we get the interval value of \[x\]. Therefore, we need to solve the inequality for \[x.\]

Complete step-by-step answer:
It is given that \[ - (x - 3) + 4 < 5 - 2x\] we are in need to solve this inequality to find the interval where the value of x exists.
Every option \[ - \infty \] exists so the left bound of $x$ is \[ - \infty \], so our need is to find the right bound of x.
Now let us solve the inequality,
\[ - (x - 3) + 4 < 5 - 2x\]
Here let us multiply the – present on the left side of the inequality,
Therefore, we get,
\[ - (x - 3) + 4 < 5 - 2x\]
\[ \Rightarrow - x + 3 + 4 < 5 - 2x\]
Now let us change the variables to one side and constants to the other side.
Hence we get the following inequality,
\[ - x + 3 + 4 < 5 - 2x\]
\[ \Rightarrow 2x - x < 5 - 3 - 4\]
Finally, we are going to solve the inequality to arrive at the right bound of x.
\[2x - x < 5 - 3 - 4\]
\[ \Rightarrow x < - 2\]
Hence we have found that the right bound of x is -2.
Initially from the given question we have that the left bound is \[ - \infty \], here the right bound of x is found to be -2.
So, the value of \[x\]lies between \[( - \infty , - 2).\]
Hence,
We have got to the conclusion that the value of x lies in the interval \[( - \infty , - 2).\]

The correct option is (C) \[( - \infty , - 2).\]

Note: While solving the inequality we have got a step of multiplying “-” in the left side, here we will not change the inequality. But if we multiply by “-” on both sides of inequality the inequality changes, that is if greater than is present it will be changed to less than. Else if less than is present it will be changed to greater than. So everyone should be careful while multiplying “-” in the inequality.