Quadratic Inequalities: Problems with Solutions

By Prof. Hernando Guzman Jaimes (University of Zulia - Maracaibo, Venezuela)
Problem 1
What is the solution to the inequality?
$x^{2}+2x-15>0$
Problem 2
Solve the inequality by factoring the expression on the left side.
$x^{2}-2x-3\leq 0$
Problem 3
$3x^{2}-x-2\leq 0$
Problem 4
Solve the inequality by factoring the expression on the left side.
$x^{2}-8x+12<0$
Problem 5
Solve the inequality by factoring the expression on the left side.
$x^{2}-5x\geq 0$
Problem 6
$3x^{2}-27<0$
Problem 7
$9x>2x^{2}-18$
Problem 8
$9x^{2}+30x>-25$
Problem 9
$4x^{2}-4x+1<0$
Problem 10
Solve the inequality by factoring the expression on the left side.
$x^{2}+6x\leq -9$

Problem 11
Solve the following inequiality
$-\left( x+1\right)\left( x+2\right) \left( x+3\right) < 0$
Problem 12
Solve the following inequiality
$-2\left( x-1\right)\left( x+\frac{1}{2}\right) \left( x-3\right) \leq 0$
Problem 13
$\left( x^{2}-1\right) \left(x^{2}-4\right) \leq 0$
Problem 14
$\left( x-1\right)^{2}\left( x+3\right) \left( x+5\right) >0$
Problem 15
$\left( x+3\right) ^{2}\left( x+4\right) \left( x-5\right)^{3}>0$
Problem 16
If $7$ times the square of a positive number is reduced by $3$ and the result is greater than $60$, what can the number be?
Problem 17
The number of diagonals $d$ of an n-sided polygon is given by the formula $d=\frac{1}{2}\left( n-1\right) n-n$.
Which polygon has the number of diagonals greater than $35$?


Problem 18
The number $t$ of dots, ordered as shown below is given by the formula $t=\frac{n(n+1)}{2}$, where n is the number of rows.
Find the range of rows if the number of dots is less than $5050$.


Problem 19
A rectangular garden should be twice as wide as it is long.
If the fenced area is greater than $98m^{2}$, what can we say about the width of the garden?
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