Solving Inequalities: A Step-by-Step Guide

by Andrew McMorgan 43 views

Hey Plastik Magazine readers! Let's dive into a common math problem: solving inequalities. Specifically, we're going to break down how to find the solution set for the inequality (4x - 3)(2x - 1) ≥ 0. Don't worry, it's not as scary as it looks! We'll go through it step by step, making sure you understand every bit. Get ready to flex those math muscles and learn something new!

Understanding Inequalities and Why They Matter

First off, what exactly is an inequality? Think of it as a mathematical statement that compares two expressions, but instead of saying they're equal (like in an equation), it says one is greater than, less than, greater than or equal to, or less than or equal to the other. In our case, (4x - 3)(2x - 1) ≥ 0 means we're looking for all the values of x that make the expression on the left side either greater than or equal to zero. Why does this matter? Well, inequalities pop up everywhere! They're used in everything from figuring out the best deal at the store (when you want to spend less than a certain amount) to modeling real-world situations in science and engineering (like ensuring a bridge can withstand more than a certain weight). Understanding inequalities is a fundamental skill that unlocks a whole world of problem-solving possibilities.

So, before we even start, let's nail down some core concepts. Inequalities tell us about the relationship between quantities, and it's super crucial to understand the direction of this relationship. Remember the symbols: > (greater than), < (less than), ≥ (greater than or equal to), and ≤ (less than or equal to). Each sign carries a specific meaning and indicates the range of values that satisfy the condition. The greater than sign suggests the values on the left are bigger than those on the right. Similarly, less than implies the opposite, meaning the left side is smaller. When we introduce the equal to, it becomes inclusive, meaning that both sides can be equal.

The Importance of the Solution Set

The solution set is the collection of all the values of x that make the inequality true. It's like finding the secret code that unlocks the solution. Finding the right solution set is the name of the game, and we'll use a strategic method to get there! It's super important, because the solution set represents all the values of x that fulfill the initial condition of the inequality. Therefore, the solution set is vital in applications such as optimization, where we seek to find the maximum or minimum values within a certain set of constraints, or in economics, where we seek to determine equilibrium points where supplies and demands are equal. It is also important in areas such as physics, where we want to determine the range of values for variables like velocity, acceleration, and position that remain valid in a given model.

Now, let's get our hands dirty and actually solve this inequality!

Step-by-Step Solution to the Inequality

Alright, let's get down to business and figure out how to solve (4x - 3)(2x - 1) ≥ 0. We'll break it down into manageable steps, so even if you're not a math whiz, you can follow along. Ready? Let's go!

Step 1: Find the Critical Points

The first step is to identify the critical points. These are the values of x that make the expression equal to zero. In other words, they're the points where the expression (4x - 3)(2x - 1) could potentially change its sign (from positive to negative or vice versa). To find them, we set each factor equal to zero and solve for x:

  • 4x - 3 = 0 => x = 3/4
  • 2x - 1 = 0 => x = 1/2

So, our critical points are x = 3/4 and x = 1/2. These are super important because they divide the number line into intervals where the expression's sign (positive or negative) remains constant.

Step 2: Create a Number Line and Test Intervals

Now, let's visualize this. Draw a number line and mark your critical points (1/2 and 3/4) on it. These points split the number line into three intervals:

  1. x < 1/2
  2. 1/2 < x < 3/4
  3. x > 3/4

Next, we're going to pick a test value (any number) within each interval and plug it into the expression (4x - 3)(2x - 1) to see if the result is positive or negative.

  • Interval 1 (x < 1/2): Let's choose x = 0. Plugging this into the expression gives us (4(0) - 3)(2(0) - 1) = (-3)(-1) = 3. Since 3 is positive, the expression is positive in this interval.
  • Interval 2 (1/2 < x < 3/4): Let's choose x = 0.6 (which is between 1/2 and 3/4). Plugging it in, we get (4(0.6) - 3)(2(0.6) - 1) = (-0.6)(0.2) = -0.12. Since -0.12 is negative, the expression is negative in this interval.
  • Interval 3 (x > 3/4): Let's choose x = 1. Plugging it in, we get (4(1) - 3)(2(1) - 1) = (1)(1) = 1. Since 1 is positive, the expression is positive in this interval.

Step 3: Determine the Solution Set

We know where the expression is positive and negative. Now, remember our original inequality: (4x - 3)(2x - 1) ≥ 0. We want to find the values of x where the expression is greater than or equal to zero. This means we're looking for the intervals where the expression is positive and the critical points themselves (where the expression equals zero).

From our testing, we know the expression is positive when x < 1/2 and when x > 3/4. The expression is equal to zero at x = 1/2 and x = 3/4. Therefore, the solution set includes:

  • All x values less than or equal to 1/2 (x ≤ 1/2)
  • All x values greater than or equal to 3/4 (x ≥ 3/4)

Step 4: Write the Solution in Interval Notation or Set Notation

We can express the solution in a few ways. Here are two common ways:

  • Interval Notation: (-∞, 1/2] ∪ [3/4, ∞)
  • Set Notation: {x | x ≤ 1/2 or x ≥ 3/4}

Both notations tell us the same thing: the solution set includes all numbers from negative infinity up to and including 1/2, and all numbers from 3/4 to positive infinity. Remember, the brackets [ ] mean that the endpoint is included, and the parentheses ( ) mean that the endpoint is not included. In our case, since the inequality includes