Solving (2x-5)(x+3) ≥ 0 A Step-by-Step Guide

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Hey guys! Today, we're diving into the world of inequalities, specifically how to solve the inequality (2x - 5)(x + 3) ≥ 0. Inequalities are a fundamental concept in algebra and calculus, and mastering them is crucial for tackling more complex problems in mathematics and physics. This comprehensive guide will walk you through the step-by-step process of solving this inequality, ensuring you understand not just the how, but also the why behind each step. Whether you're a student grappling with homework, or just someone looking to brush up on their math skills, this breakdown is for you.

Before we jump into solving our specific inequality, let's take a moment to understand what inequalities are and how they differ from equations. Unlike equations, which seek specific values that make the expressions on both sides equal, inequalities deal with ranges of values. The symbols used in inequalities include:

  • > (greater than)
  • < (less than)
  • ≥ (greater than or equal to)
  • ≤ (less than or equal to)

When we solve an inequality, we're essentially finding the set of all values that satisfy the given condition. This set can be a range of numbers, a single value, or even no values at all. In the case of (2x - 5)(x + 3) ≥ 0, we're looking for all the x values that make the expression greater than or equal to zero.

The first step in solving this inequality is to identify the critical points. These are the values of x that make the expression equal to zero. In other words, we need to solve the equation (2x - 5)(x + 3) = 0. To do this, we'll use the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. So, we set each factor equal to zero:

  • 2x - 5 = 0
  • x + 3 = 0

Solving these equations, we get:

  • 2x = 5 => x = 5/2 = 2.5
  • x = -3

These values, x = 2.5 and x = -3, are our critical points. They are crucial because they divide the number line into intervals where the expression (2x - 5)(x + 3) will have a constant sign (either positive or negative). These critical points act as boundaries, and the solution to the inequality will lie within these intervals.

A sign chart is a fantastic tool for visualizing how the expression (2x - 5)(x + 3) changes signs across different intervals. We'll create a number line and mark our critical points, -3 and 2.5, on it. These points divide the line into three intervals: (-∞, -3), (-3, 2.5), and (2.5, ∞).

     (-∞)-----(-3)-----(2.5)-----(∞)

Now, we'll pick a test value from each interval and plug it into the expression (2x - 5)(x + 3) to determine its sign in that interval. Let's choose:

  • x = -4 (from the interval (-∞, -3))
  • x = 0 (from the interval (-3, 2.5))
  • x = 3 (from the interval (2.5, ∞))

Let's evaluate:

  1. For x = -4: (2(-4) - 5)((-4) + 3) = (-8 - 5)(-1) = (-13)(-1) = 13 (Positive)
  2. For x = 0: (2(0) - 5)((0) + 3) = (-5)(3) = -15 (Negative)
  3. For x = 3: (2(3) - 5)((3) + 3) = (6 - 5)(6) = (1)(6) = 6 (Positive)

We can now fill in our sign chart:

     (-∞)----(-3)-----(2.5)-----(∞)
          +      -       +

This chart tells us that the expression (2x - 5)(x + 3) is positive in the intervals (-∞, -3) and (2.5, ∞), and negative in the interval (-3, 2.5).

Remember, we're trying to solve the inequality (2x - 5)(x + 3) ≥ 0. This means we're looking for the intervals where the expression is greater than or equal to zero. Based on our sign chart, this occurs in the intervals (-∞, -3) and (2.5, ∞). Because we have a "greater than or equal to" sign, we include the critical points in our solution.

Therefore, the solution set is:

x ∈ (-∞, -3] ∪ [2.5, ∞)

This notation means that x can be any value less than or equal to -3, or any value greater than or equal to 2.5.

To make sure we got the correct solution, it's always a good idea to verify. We can do this by picking values from our solution intervals and plugging them back into the original inequality. Let's test:

  • x = -4 (from (-∞, -3))
  • x = 3 (from (2.5, ∞))
  • x = -3 (critical point)
  • x = 2.5 (critical point)
  1. For x = -4: (2(-4) - 5)((-4) + 3) = 13 ≥ 0 (True)
  2. For x = 3: (2(3) - 5)((3) + 3) = 6 ≥ 0 (True)
  3. For x = -3: (2(-3) - 5)((-3) + 3) = (-11)(0) = 0 ≥ 0 (True)
  4. For x = 2.5: (2(2.5) - 5)((2.5) + 3) = (0)(5.5) = 0 ≥ 0 (True)

Since all test values satisfy the inequality, our solution set is correct!

Another way to solve the inequality (2x - 5)(x + 3) ≥ 0 is to expand the expression and analyze the resulting quadratic. Let's expand the expression:

(2x - 5)(x + 3) = 2x^2 + 6x - 5x - 15 = 2x^2 + x - 15

So, our inequality becomes:

2x^2 + x - 15 ≥ 0

Now, we can analyze this quadratic function. The critical points we found earlier (x = -3 and x = 2.5) are the roots of the quadratic equation 2x^2 + x - 15 = 0. The parabola opens upwards because the coefficient of the x^2 term (2) is positive. This means that the quadratic will be greater than or equal to zero outside the interval between the roots.

     Parabola opening upwards
     Roots at x = -3 and x = 2.5

Therefore, the solution is the same as before:

x ∈ (-∞, -3] ∪ [2.5, ∞)

This method provides a visual understanding of why the solution looks the way it does, especially for those who are comfortable with quadratic functions and their graphs.

When solving inequalities, it's easy to stumble upon common mistakes. Here are a few to keep in mind:

  1. Dividing or Multiplying by a Negative Number: If you divide or multiply both sides of an inequality by a negative number, you must flip the inequality sign. For example, if you have -2x > 4, dividing both sides by -2 gives x < -2, not x > -2.
  2. Forgetting to Include Critical Points: When the inequality includes “or equal to” (≥ or ≤), don't forget to include the critical points in your solution set. These points make the expression equal to zero, which satisfies the inequality.
  3. Incorrectly Interpreting the Sign Chart: Make sure you correctly interpret the sign chart. The intervals where the expression satisfies the inequality are the ones you should include in your solution.
  4. Not Verifying the Solution: Always verify your solution by plugging in test values from your solution intervals into the original inequality. This helps catch any errors you might have made.

Solving inequalities isn't just an abstract mathematical exercise; it has many practical applications in various fields. For instance:

  1. Physics: Inequalities are used to describe constraints and conditions in physical systems. For example, determining the range of velocities for a projectile to reach a certain height involves solving inequalities.
  2. Economics: In economics, inequalities are used to model budget constraints, supply and demand curves, and profit maximization problems.
  3. Engineering: Engineers use inequalities to design structures that can withstand certain loads, ensuring safety and stability.
  4. Computer Science: Inequalities are fundamental in algorithm design and analysis, especially in optimization problems and resource allocation.

Solving the inequality (2x - 5)(x + 3) ≥ 0 involves finding the critical points, creating a sign chart, and determining the solution set based on the sign chart. We found that the solution is x ∈ (-∞, -3] ∪ [2.5, ∞). Remember to verify your solution and avoid common mistakes like forgetting to flip the inequality sign when multiplying or dividing by a negative number. Understanding inequalities is a crucial skill that extends beyond mathematics and has real-world applications in various fields.

I hope this guide has helped you grasp the concept of solving inequalities. Keep practicing, and you'll become a pro in no time! Happy solving, guys!