Question

How do you solve $ - 6 \leqslant - 2x + 4 \leqslant 12 $ ?

Answer 1

Hint: In this question, we have to find the equivalent relation for the variable $ x $ . Firstly, equate the middle term with the left term and secondly equate to the right term. Then combine the two relations and find the required relation for $ x $

Complete step by step solution:
In this question, we have to find the equivalent relation for the variable $ x $ .
The given expression is $ - 6 \leqslant - 2x + 4 \leqslant 12 $
Equate the middle term which is a linear equation of $ x $ to the left-hand side term.
 $\Rightarrow - 6 \leqslant - 2x + 4 $
Take the constants on one-side.
 $ \Rightarrow - 6 - 4 \leqslant - 2x $
Or can be written as $ - 10 \leqslant - 2x $
Eliminating the negative sign,
 $\Rightarrow 2x \leqslant 10 $
Dividing by $ 2 $
 $\Rightarrow x \leqslant 5 $ $ \ldots \left( 1 \right) $
Now, equating the right-hand side term
 $ - 2x + 4 \leqslant 12 $
Take the constants on one-side.
 $\Rightarrow - 2x \leqslant 12 - 4 $ or $ - 2x \leqslant 8 $
Dividing by $ 2 $
 $\Rightarrow - x \leqslant 4 $
Eliminating the negative sign,
 $\Rightarrow - 4 \leqslant x $ $ \ldots \left( 2 \right) $
Combining the above two marked equations,
 $\Rightarrow - 4 \leqslant x \leqslant 5 $
This is our required solution.
So, the correct answer is “ $ - 4 \leqslant x \leqslant 5 $ ”.

Note: Keep in mind that when we eliminate the negative signs of the equation then the greater than equality sign will change in case of such questions. Be careful while doing the basic calculations. Here, greater and lesser signs are very important. One wrong sign can change the meaning of the whole question.