2. Steps to Solve a Quadratic Inequality

Bring the inequality to standard form: 𝑎𝑥² + 𝑏𝑥 + 𝑐 <, ≤, >, ≥ 0

In this first step, you need to rearrange the quadratic inequality so that all terms are on one side of the inequality sign. This means transforming it into a form where it's equal to zero (or a comparison to zero). For example, if you start with the inequality 2x² + 3x - 5 < 0, you would keep all terms on one side, resulting in 2x² + 3x - 5 < 0. This helps in simplifying and organizing your inequality, making it easier to analyze and graph later.

Imagine organizing a budget where you want to spend less than a certain amount. You gather all your expenses (your terms) into one list to see if it fits under that budget number (which is your standard form, or zero).

Find the roots of the quadratic by solving: 𝑎𝑥² + 𝑏𝑥 + 𝑐 = 0 Let the roots be 𝑥₁ and 𝑥₂. These divide the number line into intervals.

After rewriting the inequality, the next step is to solve the corresponding quadratic equation by setting it equal to zero. The solutions, known as the roots (denoted as x₁ and x₂), indicate where the quadratic function crosses the x-axis. These roots effectively split the number line into distinct intervals that will be analyzed to see if they satisfy the original inequality. For example, with roots x = 1 and x = 3, your intervals would be (-∞, 1), (1, 3), and (3, ∞).

Think of finding the points where a roller coaster (the quadratic) reaches the ground level (crosses the x-axis). Once you know these points, you can categorize the sections of the ride—where it's above ground and where it's below.

• Use test points or sign diagrams in each interval to determine whether the inequality is satisfied. • The parabola opens upward if 𝑎 > 0 and downward if 𝑎 < 0.

In this step, you need to evaluate each interval created from your roots to check if the quadratic inequality holds true in those regions. To do this, select a test point from each interval and substitute it back into the inequality. Analyze the sign of the result: if it's positive or negative, that tells you about the behavior of the quadratic in the entire interval. If the coefficient 'a' is positive, the parabola opens upwards, while a negative 'a' indicates a downward-opening parabola. This influences where the inequality will hold true.

Consider a chef testing the flavor of a dish at different stages of cooking (the intervals). Depending on the test sample taken, you determine if the dish is too salty or not. The sign of the test results will show if you need to add more seasoning (satisfying the inequality) or not (not satisfying the inequality).

• Use interval notation or inequality form. • Include endpoints if the inequality is non-strict (i.e., ≤ or ≥).

The final step is to clearly write down the solution to the inequality based on your sign analysis. You can express the solution using interval notation, which simplifies understanding and visual representation of where the inequality is true. If the original inequality includes non-strict signs (≤ or ≥), make sure to include the endpoints in your interval. For instance, if your test showed that the inequality holds true between the roots and includes them, you would write it as [x₁, x₂] or (x₁, x₂) based on whether they are included or not.

Imagine you are finalizing a schedule (the solution). You list the time slots when you're available (the intervals) and mark the start and end times based on whether appointments can be set up at those exact times or not. This gives a clear depiction of your availability.

--