# Solve the inequality and give your answer in interval notation: x^2 - 4 < 0 Determine the 4^{th}...

## Question:

Solve the inequality and give your answer in interval notation: {eq}x^2 - 4 < 0 {/eq}

Determine the {eq}4^{th}{/eq} term in the binomial expansion of {eq}(x^3 + 3y)^{10}.{/eq}

## Binomial Expansion:

The binominal expansion for {eq}{{\left( a+b \right)}^{n}} {/eq} is given as

{eq}\begin{align} & {{\left( a+b \right)}^{n}}=\sum\limits_{i=0}^{n}{\left( \begin{matrix} n \\ i \\ \end{matrix} \right)}{{a}^{n-i}}{{b}^{i}} \\ & ={{a}^{n}}+n{{a}^{n-1}}b+\frac{n\left( n-1 \right)}{2i}{{a}^{n-2}}{{b}^{2}}+.....+na{{b}^{n-1}}+{{b}^{n}} \\ \end{align} {/eq}

Here, {eq}\left( \begin{matrix} n \\ i \\ \end{matrix} \right)=\frac{n\left( n-1 \right)\left( n-2 \right).....\left( n-i+1 \right)}{i!},i=1,2,3,.....n {/eq}.

Binominal theorem is used in weather forecasting, ranking of applicants in exams, estimation of project cost, economic forecast etc.

The inequality is {eq}{{x}^{2}}-4<0 {/eq}

{eq}\Rightarrow \left( x-2 \right)\left( x+2 \right)<0\qquad\text{ }\!\![\!\!\text{ factorize the...

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