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

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).

Step by step expontion