Desigualdades Cuadráticas. Ejemplo 2.

Quadratic Inequalities Solutions

Steps to Solve Quadratic Inequalities

• The process for solving quadratic inequalities involves four main steps: factorization, identifying critical values, placing these values on a number line, and testing intervals to determine solutions.

• The first example presented is the inequality x^2 - x - 6 leq 0. The instructor begins by factoring the expression into binomials.

• After factoring, the expression is rewritten as (x - 3)(x + 2) leq 0. A link to a video on factorization is provided for further reference.

Identifying Critical Values

• Setting each factor equal to zero gives critical values: x = 3 and x = -2. These points are marked on a number line along with positive and negative infinity.

• The number line is segmented into three intervals based on the critical values: (-∞, -2), (-2, 3), and (3, ∞).

Testing Intervals for Solutions

• For the interval (-∞, -2), testing with x = -3:

• Calculation shows that it does not satisfy the inequality since -1 < 0.

• In the interval (-2, 3), testing with x = 0:

• Results in a valid solution as it satisfies the inequality.

• For (3, ∞), testing with x = 4:

• This also does not satisfy the inequality since it results in a positive value.

Finalizing Solution

• Evaluating at critical point x = -2:

• It satisfies the condition of being less than or equal to zero. Thus this endpoint is included in the solution set.

• Checking at x = 3:

• Also satisfies equality; hence this endpoint is included too.

• The final solution for the inequality is expressed as an interval: [-2, 3].