Solving Quadratic Inequality $x^2 - 5x + 3 < 3$ A Step-by-Step Guide

This article provides a comprehensive guide to solving the inequality $x^2 - 5x + 3 < 3$. We will break down the problem step by step, explain the underlying concepts, and illustrate the solution process with examples and graphical representations. By the end of this article, you will have a solid understanding of how to solve quadratic inequalities and interpret their solutions.

Understanding Quadratic Inequalities

Before diving into the specifics of the given inequality, let's first understand what quadratic inequalities are and how they differ from linear inequalities and equations. A quadratic inequality is a mathematical statement that compares a quadratic expression to a constant or another expression using inequality symbols such as < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to). The general form of a quadratic inequality is:

$ax^2 + bx + c < 0$ (or >, ≤, ≥), where a, b, and c are constants and a ≠ 0.

Quadratic inequalities are different from linear inequalities in that they involve a squared term ($x^2$), which gives them a parabolic shape when graphed. This parabolic nature leads to solutions that are often intervals rather than single values, as is typical with linear inequalities. When compared to quadratic equations, which seek specific values of x that make the expression equal to zero, quadratic inequalities seek ranges of x values that make the expression either less than or greater than a certain value.

When solving quadratic inequalities, we're essentially trying to find the range or ranges of x-values for which the quadratic expression satisfies the inequality. The solutions to these inequalities are intervals of the number line, representing the regions where the parabola lies above or below the x-axis, depending on the inequality sign. For instance, if we have $x^2 - 5x + 3 < 3$, we are looking for the values of x that make the quadratic expression less than 3. This contrasts with solving quadratic equations, where we find the exact points where the parabola intersects the x-axis, i.e., where the expression equals zero. Solving quadratic inequalities involves finding the intervals where the parabola lies either above or below the line defined by the constant on the other side of the inequality, which is a more nuanced problem than finding single-point solutions.

Step-by-Step Solution to $x^2 - 5x + 3 < 3$

Now, let's tackle the inequality $x^2 - 5x + 3 < 3$ step by step. To solve this inequality, we'll follow a systematic approach that includes simplifying the inequality, finding critical points, and testing intervals. This method will allow us to determine the range of x-values that satisfy the given inequality.

1. Simplify the Inequality

Our first step is to simplify the inequality by moving all terms to one side, leaving zero on the other side. This will put the inequality into a standard form that is easier to work with. In this case, we subtract 3 from both sides of the inequality:

$x^2 - 5x + 3 - 3 < 3 - 3$

This simplifies to:

$x^2 - 5x < 0$

2. Find the Critical Points

The next step is to find the critical points of the inequality. These are the points where the quadratic expression equals zero. To find them, we set the expression equal to zero and solve for x:

$x^2 - 5x = 0$

This is a quadratic equation that we can solve by factoring. We factor out an x from both terms:

$x(x - 5) = 0$

Now, we set each factor equal to zero and solve for x:

$x = 0$ or $x - 5 = 0$

So, the critical points are:

$x = 0$ and $x = 5$

These critical points are crucial because they divide the number line into intervals, within which the quadratic expression will either be positive or negative. These points are where the parabola intersects the x-axis, and the intervals between and around them are where the parabola lies either above or below the x-axis. Identifying these intervals is key to solving the inequality, as we'll determine which intervals satisfy the original condition (in this case, $x^2 - 5x < 0$).

3. Test Intervals

The critical points divide the number line into three intervals: $(-\infty, 0)$, $(0, 5)$, and $(5, \infty)$. We need to test a value from each interval in the inequality $x^2 - 5x < 0$ to determine whether the expression is negative in that interval.

• Interval $(-\infty, 0)$: Choose a test value, say $x = -1$. Substitute it into the inequality:

  $(-1)^2 - 5(-1) = 1 + 5 = 6$

  Since 6 is not less than 0, this interval does not satisfy the inequality.

• Interval $(0, 5)$: Choose a test value, say $x = 1$. Substitute it into the inequality:

  $(1)^2 - 5(1) = 1 - 5 = -4$

  Since -4 is less than 0, this interval satisfies the inequality.

• Interval $(5, \infty)$: Choose a test value, say $x = 6$. Substitute it into the inequality:

  $(6)^2 - 5(6) = 36 - 30 = 6$

  Since 6 is not less than 0, this interval does not satisfy the inequality.

4. Determine the Solution Set

Based on our interval testing, the inequality $x^2 - 5x < 0$ is satisfied only in the interval $(0, 5)$. Therefore, the solution set is:

$(0, 5)$

This means that all values of x between 0 and 5 (excluding 0 and 5 themselves) will make the original inequality $x^2 - 5x + 3 < 3$ true. Graphically, this corresponds to the portion of the parabola $y = x^2 - 5x$ that lies below the x-axis between the points x = 0 and x = 5.

Graphical Representation

The graphs of the related inequalities can provide a visual understanding of the solution. The graph of $y = x^2 - 5x + 3$ is a parabola, and the graph of $y = 3$ is a horizontal line. The solution to the inequality $x^2 - 5x + 3 < 3$ corresponds to the x-values where the parabola lies below the horizontal line. We can visualize this by plotting both equations on the same coordinate plane. The region where the parabolic curve is below the horizontal line $y = 3$ indicates the solution set for the inequality.

By plotting the graph, you will observe that the parabola intersects the line $y = 3$ at the points x = 0 and x = 5. Between these points, the parabola dips below the line, visually confirming that the solution to the inequality is indeed the interval $(0, 5)$. This graphical representation is a powerful tool for confirming algebraic solutions and for gaining an intuitive understanding of how the quadratic expression behaves in relation to the inequality.

Conclusion

The solution to the inequality $x^2 - 5x + 3 < 3$ is the interval $(0, 5)$. This means that any value of x between 0 and 5 will satisfy the inequality. We arrived at this solution by simplifying the inequality, finding critical points, testing intervals, and graphically representing the solution. Understanding these steps and the underlying concepts is crucial for solving quadratic inequalities effectively.

Solving quadratic inequalities involves a systematic approach that combines algebraic manipulation with graphical intuition. By following the steps outlined in this guide, you can confidently tackle similar problems and deepen your understanding of quadratic expressions and inequalities. The ability to solve these types of problems is crucial in various areas of mathematics and its applications, making it a valuable skill to master.