Question

How do you solve the inequality \[ - 5 < 2x + 1 < 15\]?

Answer 1

Hint: An inequality compares two values, showing if one is less than, greater than, or simply not equal to another value. Here we need to solve for ‘x’ which is a variable. Solving the given inequality is very like solving equations and we do most of the same thing but we must pay attention to the direction of inequality\[( \leqslant , > )\].

Complete step-by-step answer:
We have \[ - 5 < 2x + 1 < 15\]. That is we have two inequalities and we solve them simultaneously.
Take \[ - 5 < 2x + 1 < 15\]
Subtracting 1 on all the parts we have,
\[
\Rightarrow - 5 - 1 < 2x + 1 - 1 < 15 - 1 \\
\Rightarrow - 6 < 2x < 14 \\
 \]
We need to solve for ‘x’ but we have ‘2x’ so we divide by 2 on all the parts we have,
\[
\Rightarrow \dfrac{{ - 6}}{2} < \dfrac{{2x}}{2} < \dfrac{{14}}{2} \\
\Rightarrow - 3 < x < 7 \\
 \]
Thus the solution of \[ - 5 < 2x + 1 < 15\] is \[ - 3 < x < 7\]. In the interval form is \[( - 3,7)\].

Note: If we put the value of ‘x’ in the interval \[( - 3,7)\] in \[ - 5 < 2x + 1 < 15\] will satisfy.
Let’s take \[x = 1\] in \[ - 5 < 2x + 1 < 15\] then we have
\[
\Rightarrow - 5 < 2(1) + 1 < 15 \\
\Rightarrow - 5 < 3 < 15 \\
\]
It is correct because 3 is greater than -3 and 3 is less than 15.
We know that \[a \ne b\] says that ‘a’ is not equal to ‘b’. \[a > b\] means that ‘a’ is less than ‘b’. \[a < b\] means that ‘a’ is greater than ‘b’. These two are known as strict inequality. \[a \geqslant b\] means that ‘a’ is less than or equal to ‘b’. \[a \leqslant b\] means that ‘a’ is greater than or equal to ‘b’.

The direction of inequality do not change in these cases:
i) Add or subtract a number from both sides.
ii) Multiply or divide both sides by a positive number.
iii) Simplify a side.

The direction of the inequality change in these cases:
i) Multiply or divide both sides by a negative number.
ii) Swapping left and right hand sides.