In Exercises 69–74, solve each inequality and graph the solution set on a real number line. 2x^2 + 5x - 3 < 0

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Rewrite the inequality in standard quadratic form: \(2x^2 + 5x - 3 < 0\). This is already in standard form, so no changes are needed.

Factor the quadratic expression \(2x^2 + 5x - 3\). Look for two numbers that multiply to \(-6\) (the product of \(2\) and \(-3\)) and add to \(5\) (the middle coefficient). These numbers are \(6\) and \(-1\). Rewrite the middle term: \(2x^2 + 6x - x - 3 < 0\).

Group terms in pairs and factor each group: \((2x^2 + 6x) - (x + 3) < 0\). Factor out the greatest common factor (GCF) from each group: \(2x(x + 3) - 1(x + 3) < 0\).

Factor out the common binomial factor \((x + 3)\): \((2x - 1)(x + 3) < 0\). Now the inequality is factored.

Determine the critical points by setting each factor equal to zero: \(2x - 1 = 0\) gives \(x = \frac{1}{2}\), and \(x + 3 = 0\) gives \(x = -3\). Use these critical points to divide the number line into intervals, test each interval in the inequality \((2x - 1)(x + 3) < 0\), and identify where the product is negative. Graph the solution set on the real number line.

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주요 개념

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Inequalities

Inequalities are mathematical statements that express the relationship between two expressions that are not equal. They use symbols such as <, >, ≤, and ≥ to indicate whether one side is less than, greater than, or equal to the other. Understanding how to manipulate and solve inequalities is crucial for determining the solution set of an inequality.

Quadratic Functions

A quadratic function is a polynomial function of degree two, typically expressed in the form f(x) = ax^2 + bx + c, where a, b, and c are constants. The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the sign of 'a'. Solving inequalities involving quadratic functions often requires finding the roots of the equation and analyzing the intervals between them.

Graphing Solution Sets

Graphing solution sets involves representing the solutions of an inequality on a number line. This visual representation helps to easily identify the intervals where the inequality holds true. For quadratic inequalities, the graph typically includes open or closed circles to indicate whether endpoints are included in the solution set, based on the type of inequality used.