Question

Solve the inequality by graphing. X\(\frac{3}{4} x^2+4 x \geq 3\)

Short answer

The solution of the inequality is the x-value intervals where the graph is above the line \(y=3\). This can be found by observing the graph and marking down those x-values for which \(y \geq 3\). If you know the roots of the equation, you can also determine these intervals without a graph, as the function's value will be larger than 3 on the intervals where the function is above the x-axis and the y-value at the root is larger than 3.

Step-by-step solution

Formulate the Quadratic Equation

Rewrite the inequality to get a standard form of the quadratic equation. Subtract 3 from both sides to get: \(\frac{3}{4}x^2 + 4x - 3 = 0\)

Find the Roots

Solve the corresponding quadratic equation. Using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), we find the roots of the quadratic equation.

Draw a Rough Graph

Draw the graph of the function \(y = \frac{3}{4}x^2 + 4x - 3\). Mark the x-intercepts found in step two. The graph is a parabola opening upward since the coefficient of \(x^2\) is positive.

Interpret the Graph

Look at where the graph of the function is located above the line \(y = 3\). This is the solution region of the inequality.

Key concepts

Quadratic Equations

Quadratic equations are an essential part of algebra and appear frequently in various mathematical problems. A quadratic equation is generally of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). The highest power of the variable \( x \) is 2, which makes it a "quadratic" equation. These equations arise when working with problems involving area, projectile motion, and optimization, among others.

To solve a quadratic equation, we typically use methods such as factoring, completing the square, or applying the quadratic formula. The quadratic formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), is useful when the equation can't be easily factored. It provides the roots of the equation, which are the values of \( x \) that satisfy the equation. Understanding these roots is crucial as they help identify solutions to various problems, especially in relation to graphing.

In the given exercise, the inequality is transformed into a quadratic equation \( \frac{3}{4}x^2 + 4x - 3 = 0 \). Using the quadratic formula will help to find the roots, which are necessary for graphing the parabola.

Graphing

Graphing is a fundamental technique used to visualize equations and inequalities. It allows us to gain insights into the behavior of functions and the relationships between variables. To graph a quadratic equation, we generally look at its standard form and identify key features such as the vertex, axis of symmetry, and roots (or x-intercepts).

In the context of the exercise, graphing involves plotting the function \( y = \frac{3}{4}x^2 + 4x - 3 \) on a coordinate plane. Since the coefficient of \( x^2 \) is positive, the parabola opens upward, creating a U-shaped curve. The roots (or x-intercepts) from the quadratic equation indicate the points where the parabola crosses the x-axis, which are critical for solving inequalities.

Once the graph is drawn, the focus shifts to understanding where the function's values meet the inequality condition. In particular, we look for segments of the parabola that are positioned above or below a specific line, in this case, \( y = 3 \). This helps in determining the solution set for the inequality, demonstrating how graphing aids in interpreting mathematical situations.

Parabolas

Parabolas are the curved shapes formed by quadratic functions when graphed. They have unique properties and are symmetric around a vertical line known as the axis of symmetry. The point at which the parabola turns is called the vertex, and it can either represent the maximum or minimum value of the function, depending on the orientation of the parabola.

Understanding the orientation and position of a parabola is crucial when working with inequalities. For example, in the graph of \( y = \frac{3}{4}x^2 + 4x - 3 \), the parabola opens upward because the coefficient of the \( x^2 \) term is positive. This upward opening indicates that the vertex is the minimum point of the parabola.

In the exercise, solving the inequality by graphing involves determining where the parabola lies in relation to a horizontal line, such as \( y = 3 \). By identifying these regions, we can see where the function meets the criteria for the inequality \( \frac{3}{4}x^2 + 4x \geq 3 \). The area above this horizontal line represents the solution to the inequality, showing how parabolas visually convey solutions to quadratic inequalities.