Solving & Graphing Polynomial Inequalities: A Step-by-Step Guide

Hey math enthusiasts! Today, we're diving into the world of polynomial inequalities. Specifically, we'll tackle the inequality $5x^2 + 17x - 12 \leq 0$. We'll break down the steps to solve this, learn how to represent the solution graphically on a number line, and finally, express the solution set using interval notation. Ready to get started, guys?

Understanding Polynomial Inequalities

First things first, what exactly is a polynomial inequality? Well, it's pretty much an inequality that involves a polynomial expression. Remember those polynomial expressions you've been working with? Things like $x^2 + 2x + 1$ or $3x^3 - 4x + 7$. When we combine these expressions with inequality symbols (like $\leq$, $\geq$, $<$, or $>$), we get a polynomial inequality. Solving these inequalities means finding all the values of the variable (in this case, x) that make the inequality true. The core idea is to figure out where the polynomial is less than, greater than, or equal to zero. This often involves finding the roots (where the polynomial equals zero) and then testing intervals to see where the inequality holds true. Think of it like a puzzle where we're trying to find the specific range of x values that satisfy the condition. The goal is to isolate the x values that make the inequality true. This often starts with finding the critical points, and the sign of the polynomial.

Why Solve Polynomial Inequalities?

So, why should we care about solving these things? Well, polynomial inequalities pop up in a ton of real-world applications. They're super useful in fields like physics, engineering, and economics. For example, imagine you're designing a bridge. You might use a polynomial inequality to ensure that the structure can handle certain loads without exceeding its limits. Or, in economics, you might use one to model the profit of a company, finding the range of production levels that keep the company in the black. They are also crucial in analyzing the behavior of functions. By identifying the intervals where a function is positive or negative, or increasing or decreasing, we gain insights into its overall shape and characteristics. This is a fundamental skill in calculus and other higher-level math courses. The ability to manipulate and solve these inequalities helps to build a strong foundation of mathematical skills, enabling you to tackle more complex problems and apply those skills to various fields and real-life scenarios. It's like learning the building blocks of a bigger, more complicated structure.

Step-by-Step Solution

Let's get down to business and solve the inequality $5x^2 + 17x - 12 \leq 0$. I'll break down the process step-by-step to make it as easy as possible. Follow along, and you'll be a pro in no time.

Step 1: Find the Zeros

The first step to solve $5x^2 + 17x - 12 \leq 0$ is to find the zeros of the corresponding quadratic equation. In other words, we need to find the values of x that make $5x^2 + 17x - 12 = 0$. We can use the quadratic formula to do this. Remember the quadratic formula? It goes like this:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

In our equation, $5x^2 + 17x - 12 = 0$, we have:

• a = 5
• b = 17
• c = -12

Plug these values into the quadratic formula:

$x = \frac{-17 \pm \sqrt{17^2 - 4(5)(-12)}}{2(5)}$

$x = \frac{-17 \pm \sqrt{289 + 240}}{10}$

$x = \frac{-17 \pm \sqrt{529}}{10}$

$x = \frac{-17 \pm 23}{10}$

So, we get two possible solutions:

$x_1 = \frac{-17 + 23}{10} = \frac{6}{10} = \frac{3}{5}$

$x_2 = \frac{-17 - 23}{10} = \frac{-40}{10} = -4$

These values, $\frac{3}{5}$ and -4, are the zeros of the quadratic equation. They are the points where the parabola (the graph of the quadratic equation) intersects the x-axis.

Step 2: Create a Number Line and Test Intervals

Now that we have the zeros, we can create a number line and mark these points on it. The zeros, -4 and 3/5, divide the number line into three intervals: $(-\infty, -4)$, $(-4, \frac{3}{5})$, and $(\frac{3}{5}, \infty)$. We need to test a value from each interval to determine whether the inequality $5x^2 + 17x - 12 \leq 0$ is true or false in that interval. This means we'll choose a test value (any number) within each interval and substitute it into the original inequality. If the inequality holds true, then all values within that interval are part of the solution set.

Let's pick our test values:

• For the interval $(-\infty, -4)$, let's use -5.
• For the interval $(-4, \frac{3}{5})$, let's use 0.
• For the interval $(\frac{3}{5}, \infty)$, let's use 1.

Now, let's plug these values into the inequality $5x^2 + 17x - 12 \leq 0$ and see what happens.

• Test -5: $5(-5)^2 + 17(-5) - 12 = 5(25) - 85 - 12 = 125 - 85 - 12 = 28$. Since $28 \leq 0$ is false, the interval $(-\infty, -4)$ is not part of the solution.

• Test 0: $5(0)^2 + 17(0) - 12 = -12$. Since $-12 \leq 0$ is true, the interval $(-4, \frac{3}{5})$ is part of the solution.

• Test 1: $5(1)^2 + 17(1) - 12 = 5 + 17 - 12 = 10$. Since $10 \leq 0$ is false, the interval $(\frac{3}{5}, \infty)$ is not part of the solution.

Step 3: Determine the Solution Set

Based on our tests, the inequality $5x^2 + 17x - 12 \leq 0$ is true only for the interval $(-4, \frac{3}{5})$. Since the inequality includes