Solving 2x² + 7x - 4 < 0 A Step-by-Step Guide

Jul 13, 2025 by Admin 46 views

Solving Polynomial Inequalities: A Comprehensive Guide to 2x² + 7x - 4 < 0

Understanding Polynomial Inequalities

Polynomial inequalities, like the one we're addressing, differ from polynomial equations in that they use inequality symbols (<, >, ≤, ≥) instead of an equals sign. These inequalities define a range of values rather than specific solutions, making their solutions a set of intervals on the real number line. To solve a polynomial inequality, we aim to identify these intervals where the polynomial expression satisfies the given inequality. This process typically involves factoring the polynomial, finding critical values, and then testing intervals to determine where the inequality holds true.

In our specific case, we are tasked with solving the quadratic inequality 2x² + 7x - 4 < 0. This means we need to find all the values of 'x' for which the quadratic expression 2x² + 7x - 4 results in a negative value. The steps to solve this involve factoring the quadratic, identifying the critical points (where the expression equals zero), and testing the intervals defined by these points to see where the expression is indeed less than zero. By doing this, we can accurately determine the solution set for the inequality.

Polynomial inequalities are crucial in various mathematical and real-world contexts. They frequently appear in calculus when determining the intervals where a function increases or decreases and in optimization problems where we seek to maximize or minimize quantities within constraints. In practical applications, polynomial inequalities help model scenarios such as determining the range of production levels that yield a profit, or the acceptable temperature range for a chemical reaction. Therefore, understanding how to solve these inequalities is not just an academic exercise but a valuable skill for problem-solving in numerous fields.

Step 1: Factor the Polynomial

The first crucial step in solving the polynomial inequality 2x² + 7x - 4 < 0 is to factor the quadratic expression. Factoring allows us to rewrite the quadratic in a form that makes it easier to identify the values of 'x' that make the expression equal to zero, which are the critical points for our inequality.

To factor 2x² + 7x - 4, we look for two binomials that, when multiplied together, give us the original quadratic. This involves finding two numbers that multiply to give the product of the leading coefficient (2) and the constant term (-4), which is -8, and add up to the middle coefficient (7). The numbers 8 and -1 satisfy these conditions, since 8 * -1 = -8 and 8 + (-1) = 7. Using these numbers, we can rewrite the middle term of the quadratic and then factor by grouping:

2x² + 7x - 4 = 2x² + 8x - x - 4

Now, we group the terms and factor out the greatest common factor (GCF) from each group:

= 2x(x + 4) - 1(x + 4)

We can see that (x + 4) is a common factor, so we factor it out:

= (2x - 1)(x + 4)

Thus, the factored form of the quadratic expression 2x² + 7x - 4 is (2x - 1)(x + 4). This factorization is a critical step because it transforms the inequality into a product of two factors, which is easier to analyze. The factored form allows us to identify the values of 'x' that make the expression equal to zero, which are the points where the polynomial may change sign.

Step 2: Find the Critical Values

Once we have factored the polynomial, the next step is to find the critical values. These values are the solutions to the equation (2x - 1)(x + 4) = 0. Critical values are essential because they are the points where the polynomial expression can change its sign (from positive to negative or vice versa). They essentially divide the real number line into intervals that we will then test to solve the inequality.

To find the critical values, we set each factor equal to zero and solve for 'x':

2x - 1 = 0 Solving for 'x', we get: 2x = 1 x = 1/2
x + 4 = 0 Solving for 'x', we get: x = -4

Therefore, the critical values are x = 1/2 and x = -4. These values are the x-intercepts of the quadratic function y = 2x² + 7x - 4. On the real number line, these points are where the parabola intersects the x-axis. The critical values divide the number line into three intervals: (-∞, -4), (-4, 1/2), and (1/2, ∞). The sign of the polynomial expression 2x² + 7x - 4 will be constant within each of these intervals.

Understanding the importance of critical values is vital for solving polynomial inequalities. They serve as boundary points that separate regions where the inequality is either true or false. By identifying these values, we can narrow down the possible solution set to specific intervals, making the process of solving the inequality more manageable. In the following steps, we will use these critical values to test the intervals and determine which ones satisfy the original inequality.

Step 3: Test Intervals

With the critical values identified, the next step in solving the polynomial inequality 2x² + 7x - 4 < 0 is to test the intervals that these values create on the real number line. The critical values, which we found to be x = -4 and x = 1/2, divide the number line into three distinct intervals: (-∞, -4), (-4, 1/2), and (1/2, ∞). To determine whether the inequality is satisfied in each interval, we select a test point within each interval and substitute it into the factored form of the inequality, which is (2x - 1)(x + 4) < 0.

Here's how we test each interval:

Interval (-∞, -4): Choose a test point, say x = -5. Substitute it into the inequality: (2(-5) - 1)(-5 + 4) = (-11)(-1) = 11 Since 11 is not less than 0, the inequality is not satisfied in this interval.
Interval (-4, 1/2): Choose a test point, say x = 0. Substitute it into the inequality: (2(0) - 1)(0 + 4) = (-1)(4) = -4 Since -4 is less than 0, the inequality is satisfied in this interval.
Interval (1/2, ∞): Choose a test point, say x = 1. Substitute it into the inequality: (2(1) - 1)(1 + 4) = (1)(5) = 5 Since 5 is not less than 0, the inequality is not satisfied in this interval.

By testing these intervals, we have determined that the inequality 2x² + 7x - 4 < 0 is satisfied only in the interval (-4, 1/2). This means that the solutions to the inequality lie between -4 and 1/2, but do not include the endpoints themselves because the inequality is strict (i.e., '<' rather than '≤').

This interval testing method is crucial for solving polynomial inequalities because it provides a clear and systematic way to determine the range of values that satisfy the given condition. The use of test points simplifies the process, allowing us to quickly assess the sign of the polynomial expression in different regions of the number line. In the following step, we will express this solution set in interval notation and graph it on a real number line.

Step 4: Express the Solution Set

After testing the intervals, we know that the solution to the polynomial inequality 2x² + 7x - 4 < 0 lies within the interval (-4, 1/2). Now, we need to express this solution set in interval notation and graphically represent it on a real number line. Interval notation is a concise way to represent a set of numbers using endpoints and parentheses or brackets. Graphing the solution set visually reinforces our understanding of the solution.

Interval Notation

The solution set for 2x² + 7x - 4 < 0 in interval notation is (-4, 1/2). This notation indicates that the solution includes all real numbers between -4 and 1/2, but it does not include the endpoints -4 and 1/2 themselves. The parentheses '(' and ')' are used because the inequality is strict (i.e., '<'), meaning the values at the endpoints do not satisfy the inequality.

If the inequality had been 2x² + 7x - 4 ≤ 0, the interval notation would have been [-4, 1/2], using square brackets '[' and ']' to indicate that the endpoints are included in the solution set.

Graphing the Solution Set

To graph the solution set on a real number line, we first draw a number line and mark the critical values -4 and 1/2. Since the inequality is strict, we use open circles at these points to indicate that they are not included in the solution. Then, we shade the region between -4 and 1/2, representing the interval where the inequality holds true.

<-----------------|---------|----------------->
                  o=========o
         -4       1/2

In the graph above, the open circles at -4 and 1/2 signify that these points are not part of the solution. The shaded region between them visually represents all the real numbers that satisfy the inequality 2x² + 7x - 4 < 0. This graphical representation provides a clear and intuitive understanding of the solution set.

Expressing the solution set in both interval notation and graphical form is crucial for a complete understanding of the solution to the polynomial inequality. Interval notation provides a compact and precise way to communicate the solution, while the graph offers a visual representation that can aid in comprehension and problem-solving.

Conclusion

In conclusion, solving the polynomial inequality 2x² + 7x - 4 < 0 involves several key steps: factoring the polynomial, finding the critical values, testing intervals, and expressing the solution set in interval notation and graphically. By following these steps, we can systematically determine the range of values that satisfy the inequality.

First, we factored the quadratic expression 2x² + 7x - 4 into (2x - 1)(x + 4). This factorization allowed us to identify the critical values, which are the points where the polynomial equals zero. The critical values, x = -4 and x = 1/2, divide the real number line into intervals.

Next, we tested each interval to determine where the inequality 2x² + 7x - 4 < 0 holds true. By choosing a test point in each interval and substituting it into the factored inequality, we found that the inequality is satisfied in the interval (-4, 1/2).

Finally, we expressed the solution set in interval notation as (-4, 1/2) and graphically represented it on a real number line. The interval notation concisely conveys the range of solutions, while the graph provides a visual representation that enhances understanding.

Understanding how to solve polynomial inequalities is a fundamental skill in algebra and calculus. It has applications in various fields, including optimization problems, determining intervals of increase and decrease for functions, and modeling real-world scenarios. The systematic approach outlined in this guide provides a solid foundation for tackling a wide range of polynomial inequalities. By mastering these techniques, students and practitioners can confidently solve these problems and apply the concepts to practical situations.