Question

Solve the inequality by graphing. \(3 x^2+5 x-3<1\)

Short answer

The solutions to the inequality \(3x^2 + 5x - 3 < 1\) are the shaded areas on the x-axis obtained in the final step, which represent the intervals of the variable \(x\) that satisfy the inequality.

Step-by-step solution

Rewrite the inequality into standard form

To do this, subtract 1 from both sides of the inequality to get it into standard form. This results in the inequality \(3x^2 + 5x - 4 < 0\).

Solve the associated quadratic equation

Set \(3x^2 + 5x - 4 = 0\) and solve this quadratic equation to get the roots of the equation, which are the points where the graph of the equation touches or crosses the x-axis. This can be done using the quadratic formula: \(x = [-b ± sqrt(b^2 - 4ac)] / (2a)\). Substituting \(a=3\), \(b=5\), and \(c=-4\) into the formula provides the solutions to the equation.

Graph the equation and check intervals

Draw the graph of the equation \(y = 3x^2 + 5x - 4\), with the roots obtained in the previous step used as the x-intercepts. The x-axis is divided into several intervals by these x-intercepts. Choose a test point in each interval and substitute it into the original inequality. If the inequality holds, shade the respective interval on the x-axis. The solution to the inequality is the shaded area on the x-axis.

Key concepts

Graphing Inequalities

Graphing inequalities involving quadratic functions provides a visual representation of the solutions to the inequality. To start, you first need to identify the roots of the associated quadratic equation. Once the roots are determined, these roots, or x-intercepts, will section the x-axis into intervals. Visualizing the graph, you create a parabola that will either open upwards or downwards depending on the coefficient of the squared term.

To determine if an interval is part of the solution for the original inequality, select a test point from within that interval and substitute it into your inequality. If the resulting statement is true, it indicates that the entire interval satisfies the inequality. You then shade this region to illustrate that all points within are solutions. The graph for our example inequality, let's say, will show a parabola opening upwards, with the shaded regions indicating where the graph lies below the x-axis.

Quadratic Formula

The quadratic formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), is a reliable method to find the roots of a quadratic equation \( ax^2 + bx + c = 0 \). It allows us to solve for 'x' by substituting the coefficients from the equation directly into the formula. The discriminant, \( b^2 -4ac \), determines the nature of the roots; if it is positive, there are two real and distinct roots, if it is zero, there is one real root, and if it is negative, there are no real roots. For the exercise in question, applying the quadratic formula results in obtaining the critical points that section the x-axis into intervals which are then tested for solutions to the inequality.

Standard Form of an Inequality

The standard form of an inequality is akin to that of a quadratic equation, typically represented as \( ax^2 + bx + c < 0 \) or \( ax^2 + bx + c > 0 \), where 'a', 'b', and 'c' are real numbers, and 'a' is not zero. The standard form is essential for analyzing the inequality because it places the quadratic function on one side and zero on the other, making it easier to identify the coefficients for the quadratic formula and to graph the inequality. When solving the given exercise, converting the inequality to standard form by subtracting 1 from both sides allowed us to properly apply the quadratic formula and to graph the inequality.

Quadratic Equation Roots

The roots of a quadratic equation, found from the equation set to zero, indicate where the function intersects the x-axis. These roots are crucial for solving quadratic inequalities, as they partition the number line into intervals that can be tested to determine where the inequality holds true. There can be two roots, one root, or no real roots, depending on the discriminant of the quadratic formula. For the inequality \( 3x^2 + 5x - 4 < 0 \), using the quadratic formula gives us the roots that are used as boundary points. These boundary points serve as critical markers for graphing the inequality and are the building blocks for identifying the solution set to the inequality.