Name: Desigualdades Cuadráticas. Ejemplo 1.
Uploaded: 2020-05-06T00:47:22.000Z
Duration: 18 min 49 s
Description: Con este video aprenderás a resolver desigualdades cuadráticas y a analizar el conjunto solución a partir de la solución gráfica y con lo cual podrás determinar la solución en su forma de intervalo.

Desigualdades Cuadráticas. Ejemplo 1.

Quadratic Inequalities Explained

Introduction to Quadratic Inequalities

The video introduces quadratic inequalities, specifically focusing on the expression x^2 - 7x - 10 > 0 and the method of factorization using the "scissors method."

A link to a previous video explaining the scissors method is provided for viewers unfamiliar with it.

Factorization Process

The instructor identifies pairs of numbers that multiply to -10 (the constant term), such as 2 times -5 or -1 times 10. They choose 2 and -5 based on their signs.

Cross-multiplication is performed to verify that the chosen factors are correct, confirming that they yield the original quadratic expression.

Identifying Critical Values

After factorization, each factor is set equal to zero: x + 2 = 0 and x - 5 = 0, leading to critical values of x = -2 and x = 5. These values are plotted on a number line.

The number line is segmented into three intervals: (-∞, -2), (-2, 5), and (5, ∞). Each interval will be tested for solutions.

Testing Intervals for Solutions

A test value from each interval is substituted back into the original inequality:

For interval (-∞, -2), testing with x = 0: results in a true statement (10 > 0). Thus this interval contributes to the solution.

For interval (-2, 5), testing with x = 3: results in a false statement (-2 > 0). This means this interval does not contribute to the solution.

For interval (5, ∞), testing with x = 6: results in a true statement (4 > 0). Hence this interval also contributes to the solution.

Determining Inclusion of Critical Points

To determine if critical points (-2 and +5) are included in the solution set:

Testing at x = -2: yields false (0 > 0), indicating it’s not included (open circle).

Testing at x = +5: also yields false (0 > 0), confirming it’s not included either (open circle).

Final Solution Representation

The final solution includes all valid intervals: from negative infinity up to but not including -2, and from just above +5, extending towards positive infinity.

Expressed in interval notation:

Solution: (-∞, -2) ∪ (5, ∞).

This indicates both endpoints are excluded from contributing solutions.