Question

2 in 1

1) Determine the number and type of solutions for the equation.
2) Solve the inequality, write in interval notation.

Answer 1

1)
2x^2 - 13x - 24 = 0;
the discriminant is : ( - 13 )^2 - 4 * 2 * ( -24 ) = 169 + 192 = 361 = 19^2 => we have two different rational-number solutions ;

2)
[ -2( x + 2 ) - 3( x - 5 ) ] / [ ( x - 5 )( x + 2 ) ] < 0 <=>

( -5x + 11 ) / [ ( x - 5 )( x + 2 ) ] < 0

We have 2 situations :
a) - 5x + 11 < 0 and ( x - 5 )( x + 2 ) > 0 => x∈ ( 11 / 5 , + oo ) and x∈( -oo, - 2 )U
( 5 , + oo ) => x∈( 5, +oo);
b) - 5x + 11 > 0 and ( x - 5 )( x + 2 ) < 0 => x∈(-oo, 11/5) and x∈( -2, 5 ) =>
x∈( -2, 11/5 );

Finally, x∈ U (-2, 11 / 5 ) U ( 5, +oo).