2 in 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).

Answered by crisforp | 2024-06-10

The equation has two distinct rational-number solutions. The inequality solution is in the intervals (-2, 11/5) and (5, +∞). Thus, the final solution in interval notation is x ∈ (-2, 11/5) ∪ (5, +∞).
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Answered by crisforp | 2025-06-16