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[Solved] Solving the given inequality

 
Inequalities
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June 15, 2021 12:24 pm
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(@armor)
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Solve the inequality \sqrt{2x+5}>x+1.

 

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June 15, 2021 12:24 pm
Topic starter
Staff 
(@armor)
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To make the square root valid, we need \small 2x+5\geq 0\Leftrightarrow x\geq -\frac{5}{2}.

When \small x+1<0,  i.e.  \small x<-1,  we have \small \sqrt{2x+5}\geq 0>x+1,  thus the original inequality has the solution \small -\frac{5}{2}\leq x<-1.

When \small x\geq -1,  the original inequality has the solution \small -1\leq x<2.

The union of these two solution sets provides the solution of the original inequality:

\small \left \{ x|-\frac{5}{2}\leq x<2 \right \}.

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