Solving & Graphing: -4(x+3) ≤ -2-2x Inequality

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Have you ever encountered an inequality and wondered how to visually represent its solution set? Inequalities, unlike equations, often have a range of solutions rather than a single value. A number line is a powerful tool for illustrating these solutions. In this comprehensive guide, we'll break down the process of solving the inequality -4(x+3) ≤ -2-2x and accurately depicting its solution set on a number line. Let's dive in and conquer the world of inequalities together!

Understanding Inequalities and Number Lines

Before we jump into the specifics of our example, let's build a solid foundation by understanding what inequalities are and how number lines work. Inequalities are mathematical statements that compare two expressions using symbols like < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). Unlike equations, which have a single solution (or a finite set of solutions), inequalities typically have a range of values that satisfy the statement. This is where the number line comes in handy.

A number line is a visual representation of all real numbers, extending infinitely in both positive and negative directions. It provides a clear way to visualize the solutions to an inequality. Each point on the line corresponds to a real number, and we can use different notations, such as open and closed circles or shaded regions, to indicate which values are included in the solution set. Grasping these fundamental concepts is crucial for effectively solving and graphing inequalities.

Step-by-Step Solution of the Inequality -4(x+3) ≤ -2-2x

Now, let's tackle our specific inequality: -4(x+3) ≤ -2-2x. We'll break down the solution process into clear, manageable steps.

1. Distribute and Simplify

The first step is to simplify both sides of the inequality by applying the distributive property. This means multiplying the -4 by both terms inside the parentheses:

-4 * x + (-4) * 3 ≤ -2 - 2x

This simplifies to:

-4x - 12 ≤ -2 - 2x

2. Isolate the Variable Term

Our next goal is to gather all the terms containing 'x' on one side of the inequality. To do this, we can add 2x to both sides:

-4x - 12 + 2x ≤ -2 - 2x + 2x

This gives us:

-2x - 12 ≤ -2

3. Isolate the Constant Term

Now, let's isolate the constant terms on the other side of the inequality. We can achieve this by adding 12 to both sides:

-2x - 12 + 12 ≤ -2 + 12

Simplifying, we get:

-2x ≤ 10

4. Solve for x

To finally solve for 'x', we need to divide both sides of the inequality by -2. Here's a crucial point: when we multiply or divide an inequality by a negative number, we must reverse the direction of the inequality sign. So, we have:

(-2x) / -2 ≥ 10 / -2

This results in:

x ≥ -5

Therefore, the solution to the inequality is x is greater than or equal to -5.

Representing the Solution on a Number Line

With the solution in hand, we can now visually represent it on a number line. This representation provides a clear picture of all the values that satisfy the inequality. This is a crucial step to make sure we fully understand the solution. Visualizing it helps us solidify our understanding.

1. Draw a Number Line

Start by drawing a horizontal line. Mark zero (0) in the middle and then mark a few integers to the left and right, ensuring you include the key value from our solution, which is -5. The number line should extend far enough in both directions to clearly represent the solution.

2. Locate the Key Value

Find -5 on your number line. This is the boundary point for our solution set.

3. Use the Correct Notation

Since our solution includes 'x is greater than or equal to -5' (x ≥ -5), we use a closed circle (or a filled-in circle) at -5 on the number line. A closed circle indicates that -5 is included in the solution set. If the inequality was 'x > -5' (greater than only), we would use an open circle to show that -5 is not included.

4. Shade the Solution Region

The inequality x ≥ -5 means that all values greater than or equal to -5 are solutions. To represent this, we shade the region of the number line to the right of -5, extending towards positive infinity. This shaded region visually represents all the numbers that satisfy the inequality.

The Completed Number Line

The number line should now have a closed circle at -5 and a shaded region extending to the right. This visual representation clearly illustrates that the solution set includes -5 and all numbers greater than -5.

Common Mistakes to Avoid

Working with inequalities can sometimes be tricky, and it's easy to make mistakes if you're not careful. Here are a few common errors to watch out for:

  • Forgetting to Reverse the Inequality Sign: As we discussed earlier, you must reverse the inequality sign when multiplying or dividing both sides by a negative number. This is a critical step, and overlooking it will lead to an incorrect solution.
  • Incorrectly Interpreting the Inequality Symbol: Be sure you understand the difference between 'less than' (<) and 'less than or equal to' (≤), and similarly for 'greater than' (>) and 'greater than or equal to' (≥). Using the wrong symbol will result in an inaccurate solution and number line representation.
  • Using the Wrong Circle Notation: Remember, a closed circle indicates that the endpoint is included in the solution set, while an open circle indicates that it is not. Using the wrong notation will misrepresent the solution.
  • Shading the Incorrect Region: Make sure you shade the correct region of the number line based on the direction of the inequality. For 'greater than' inequalities, shade to the right; for 'less than' inequalities, shade to the left.

By being mindful of these common pitfalls, you can increase your accuracy and confidence when solving and graphing inequalities.

Examples and Practice Problems

To solidify your understanding, let's explore a few more examples and practice problems.

Example 1: Solve and graph the inequality 2x + 3 > 7.

  1. Subtract 3 from both sides: 2x > 4
  2. Divide both sides by 2: x > 2
  3. On the number line, place an open circle at 2 (since it's 'greater than', not 'greater than or equal to').
  4. Shade the region to the right of 2.

Example 2: Solve and graph the inequality -3x + 1 ≤ 10.

  1. Subtract 1 from both sides: -3x ≤ 9
  2. Divide both sides by -3 (and reverse the inequality sign): x ≥ -3
  3. On the number line, place a closed circle at -3 (since it's 'greater than or equal to').
  4. Shade the region to the right of -3.

Now, try these practice problems on your own:

  1. Solve and graph: 4x - 5 < 3
  2. Solve and graph: -2x + 6 ≥ 2

Working through these examples and practice problems will reinforce your skills and deepen your understanding of solving and graphing inequalities.

Real-World Applications of Inequalities

Inequalities aren't just abstract mathematical concepts; they have numerous real-world applications. Understanding how to solve and graph inequalities can be valuable in various situations. Let’s explore a few scenarios where inequalities come into play.

  • Budgeting: Imagine you have a budget for groceries. You can use an inequality to represent the amount of money you can spend. For example, if you have $100 to spend, the inequality would be: total cost of groceries ≤ $100. You can then use this inequality to determine how much of each item you can buy.
  • Speed Limits: Speed limits on roads are expressed as inequalities. For instance, a speed limit of 65 mph means that your speed must be less than or equal to 65 mph. This can be written as: speed ≤ 65 mph.
  • Age Restrictions: Many activities have age restrictions that can be represented using inequalities. For example, if you need to be at least 16 years old to get a driver's license, this can be written as: age ≥ 16.
  • Temperature Ranges: Weather forecasts often provide temperature ranges. For example, a forecast might predict a high temperature of no more than 80°F. This can be expressed as: temperature ≤ 80°F.
  • Manufacturing Tolerances: In manufacturing, products often have tolerances, which are acceptable ranges of variation in their dimensions. These tolerances can be represented using inequalities. For example, if a part should be 10 cm long with a tolerance of ±0.1 cm, the length can be represented as: 9.9 cm ≤ length ≤ 10.1 cm.

By recognizing these real-world applications, you can appreciate the practical significance of inequalities and their role in everyday life.

Conclusion

Mastering the art of solving and graphing inequalities is a fundamental skill in mathematics. In this guide, we've walked through the step-by-step process of solving the inequality -4(x+3) ≤ -2-2x and representing its solution set on a number line. We've also highlighted common mistakes to avoid and explored real-world applications of inequalities. By understanding the concepts and practicing regularly, you can confidently tackle any inequality problem that comes your way. Remember, the number line is your friend when it comes to visualizing solutions and ensuring accuracy. So, keep practicing, keep exploring, and keep conquering the world of mathematics!

For further learning and practice, consider visiting resources like Khan Academy's Algebra I section, which offers comprehensive lessons and exercises on inequalities and other algebra topics.