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Question

# Solve the absolute value inequality to find the interval of convergence.$\left| \dfrac{x+1}{2} \right| < 1$

Solution

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Solve the given inequality as shown below, follow the steps:

$\Big|\dfrac{x+1}{2}\Big| \text{\textless} 1\qquad$ [recall: $|a|\text{\textless} b\Rightarrow -b\text{\textless} a\text{\textless} b$]

$\Rightarrow -1\text{\textless} \dfrac{x+1}{2}\text{\textless} 1\qquad$ [multiply each side by $2$]

$\Rightarrow -2\text{\textless} x+1\text{\textless} 2\qquad$ [subtract $1$ in each side]

$\Rightarrow -2-1\text{\textless} x\text{\textless} 2-1$

$\Rightarrow -3\text{\textless} x\text{\textless} 1$

The solution written in interval notation is: $\quad x\in (-3,1)$

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