Solving Inequalities: A Step-by-Step Guide

by Dimemap Team 43 views

Hey math enthusiasts! Let's dive into the world of inequalities and figure out the solution set for the inequality (4x - 3)(2x - 1) ≥ 0. Understanding how to solve these types of problems is super important in algebra, and I'm here to break it down for you in a way that's easy to follow. We'll explore the critical points, test intervals, and arrive at the correct solution set, making sure you're well-equipped to tackle similar problems. So, let's get started, shall we?

Understanding the Basics of Inequalities

Before we jump into the specific problem, let's quickly recap what inequalities are all about. Inequalities are mathematical statements that compare two values, indicating that one is greater than, less than, greater than or equal to, or less than or equal to the other. Unlike equations, which have a single solution (or a set of discrete solutions), inequalities often have a range of solutions. This means the solution set can include all values within a certain interval. In our case, the inequality involves a product of two binomials, (4x - 3) and (2x - 1), and we're looking for the values of x that make this product greater than or equal to zero. This setup tells us we need to find the values of x for which the product is either positive or equal to zero.

To solve this, we'll use a strategic approach that involves finding critical points, testing intervals, and determining the solution set. The core idea is that the sign of the expression (4x - 3)(2x - 1) can only change at the points where either (4x - 3) = 0 or (2x - 1) = 0. These points divide the number line into intervals, within which the expression maintains a consistent sign (either positive or negative). We then test these intervals to determine which ones satisfy the inequality.

Step-by-Step Solution: Finding the Solution Set

Let's get down to the nitty-gritty and find the solution set for (4x - 3)(2x - 1) ≥ 0. Here’s how we'll do it:

  1. Find the critical points: These are the values of x that make each factor equal to zero. To find them, set each factor to zero and solve for x.

    • For (4x - 3) = 0, solve for x: 4x - 3 = 0 4x = 3 x = 3/4
    • For (2x - 1) = 0, solve for x: 2x - 1 = 0 2x = 1 x = 1/2 So, our critical points are x = 3/4 and x = 1/2.

  2. Plot the critical points on a number line: This helps visualize the intervals we need to test. Draw a number line and mark 1/2 and 3/4 on it. These points divide the number line into three intervals: (-∞, 1/2), (1/2, 3/4), and (3/4, ∞).

  3. Test the intervals: Choose a test value from each interval and plug it into the expression (4x - 3)(2x - 1) to see if the result is greater than or equal to zero.

    • Interval (-∞, 1/2): Let's test x = 0. Then, (4(0) - 3)(2(0) - 1) = (-3)(-1) = 3. Since 3 ≥ 0, this interval satisfies the inequality.
    • Interval (1/2, 3/4): Let's test x = 0.6. Then, (4(0.6) - 3)(2(0.6) - 1) = (2.4 - 3)(1.2 - 1) = (-0.6)(0.2) = -0.12. Since -0.12 < 0, this interval does not satisfy the inequality.
    • Interval (3/4, ∞): Let's test x = 1. Then, (4(1) - 3)(2(1) - 1) = (1)(1) = 1. Since 1 ≥ 0, this interval satisfies the inequality.

  4. Determine the solution set: Include the intervals where the test values satisfied the inequality (i.e., the result was ≥ 0). Also, include the critical points themselves because the inequality includes “equal to.”

    • The intervals that satisfy the inequality are (-∞, 1/2) and (3/4, ∞). Include the critical points 1/2 and 3/4. The solution set is therefore all values of x less than or equal to 1/2 and all values of x greater than or equal to 3/4.

So, the solution set is {x | x ≤ 1/2 or x ≥ 3/4}. This means any value of x in these intervals will make the original inequality true.

Graphing the Solution Set

Graphing the solution set helps visualize the values that satisfy the inequality. On a number line:

  • Draw a closed circle at x = 1/2 to show that it's included in the solution.
  • Shade the number line to the left of 1/2, indicating all values less than or equal to 1/2.
  • Draw a closed circle at x = 3/4 to show that it's included in the solution.
  • Shade the number line to the right of 3/4, indicating all values greater than or equal to 3/4.

This graph clearly illustrates the range of x values that make the inequality true.

Common Mistakes to Avoid

When solving inequalities like these, there are a few common pitfalls to watch out for. Avoiding these mistakes will help you arrive at the correct solution more consistently:

  • Forgetting to Test Intervals: Always test values in each interval created by the critical points. This step is crucial for determining where the inequality holds true. Skipping this might lead to including or excluding the wrong intervals.

  • Not Including Critical Points: Be careful about whether to include the critical points in your solution set. If the inequality includes