Does the following system of inequalities have a solution. 3*x^2+6*x > 9 and 3 > x^2+2*x

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The solution of the system of inequalities 3*x^2+6*x > 9 and 3 > x^2+2*x has to be determined.

First solve the inequality : 3*x^2+6*x > 9

=> x^2 + 2x > 3

=> x^2 + 2x - 3 > 0

=> x^2 + 3x - x - 3 >0

=> x(x + 3) - 1(x + 3) >0

=> (x - 1)(x + 3) > 0

This is true if:

  • x - 1 > 0 and x + 3 > 0

=> x > 1 and x > -3

=> x > 1

  • x - 1 < 0 and x + 3 < 0

=> x < 1 and x < -3

=> x < -3

The solution of the inequality is `(-oo, -3)U(1, oo)`

Now solve the inequality 3 > x^2+2*x. 

3 > x^2+2*x

=> 0 > x^2 + 2x - 3

=> 0 > x^2 + 3x - x - 3

=> 0 > x(x + 3) - 1(x + 3)

=> 0 > (x - 1)(x + 3)

This holds if:

  • x - 1 > 0 and x + 3 < 0

=> x > 1 and x < -3

There are no values of x that satisfy both the conditions

  • x - 1 < 0 and x + 3 > 0

=> x < 1 and x > -3

The solution of the inequality is (-3 , 1)

The intersection of (-3, 1) and `(-oo, -3)U(1, oo)` is a null set.

There is no value of x that satisfies both the inequalities.

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