Question

How do you solve \[5 < 2x+7 < 13\] ?

Answer 1

Hint: First try to reduce the term containing variable by different arithmetic operations like addition, subtraction, multiplication and division to separate the variable from the constants. While performing an operation, apply it on every terms of the inequality.

Complete step by step answer:
While solving this set of inequalities we need to perform each operation to all three parts of the set of inequalities to keep everything balanced.
Subtracting 7 from each term of the inequalities, we get
\[\begin{align}
  & 5-7 < \left( 2x+7 \right)-7 < 13-7 \\
 & \Rightarrow -2 < 2x < 6 \\
\end{align}\]
Now dividing each term of the inequalities by 2, we get
\[\dfrac{-2}{2} < \dfrac{2x}{2} < \dfrac{6}{2}\]
Cancelling out 2 both from the numerator and the denominator of the middle term and reducing the left term as $\dfrac{-2}{2}=-1$ and the right term as $\dfrac{6}{2}=3$, we get
\[\Rightarrow -1 < x < 3\]
Therefore we can conclude that $x\in \left( -1,3 \right)$.
This is the required solution of the given question.

Note:
Separating the constants and the variables should be the first approach for solving this question. Necessary simplification and calculation should be done for solving the inequality. The range of ‘x’ can be determined by solving the inequality as $x\in \left( -1,3 \right)$.