Graphing Inequality (1/3)(12-4x) ≤ 16: A Visual Guide
Let's dive into the world of inequalities and learn how to represent their solutions graphically. In this article, we'll specifically focus on the inequality (1/3)(12-4x) ≤ 16. We will break down the steps to solve this inequality and then illustrate how to represent the solution set on a graph. So, grab your pencils and paper, and let's get started!
Understanding Inequalities
Before we jump into solving our specific inequality, it's crucial to understand what inequalities are and how they differ from equations. 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 often have a range of solutions.
Key Concepts to Remember:
- Less Than (<): One value is smaller than the other.
- Greater Than (>): One value is larger than the other.
- Less Than or Equal To (≤): One value is smaller than or equal to the other.
- Greater Than or Equal To (≥): One value is larger than or equal to the other.
When graphing inequalities, these symbols dictate how we represent the solution set on a number line. We'll use open circles for strict inequalities (< and >) and closed circles for inclusive inequalities (≤ and ≥).
Solving the Inequality (1/3)(12-4x) ≤ 16
Now, let's tackle the inequality (1/3)(12-4x) ≤ 16 step by step. Solving an inequality involves isolating the variable (in this case, x) on one side of the inequality sign. Here’s how we do it:
Step 1: Distribute the (1/3)
First, we distribute the (1/3) across the terms inside the parentheses:
(1/3) * 12 - (1/3) * 4x ≤ 16
This simplifies to:
4 - (4/3)x ≤ 16
Step 2: Isolate the Term with x
Next, we want to isolate the term with x. To do this, subtract 4 from both sides of the inequality:
4 - (4/3)x - 4 ≤ 16 - 4
Which gives us:
-(4/3)x ≤ 12
Step 3: Solve for x
Now, we need to get x by itself. To do this, we'll multiply both sides of the inequality by -3/4. Important Note: When you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign.
(-3/4) * -(4/3)x ≥ 12 * (-3/4)
This simplifies to:
x ≥ -9
So, the solution to the inequality is x is greater than or equal to -9.
Graphing the Solution Set
Now that we've solved the inequality, let's represent the solution set on a number line. This visual representation helps us understand all the possible values of x that satisfy the inequality x ≥ -9.
Step 1: Draw a Number Line
Start by drawing a horizontal line. This is our number line. Mark zero (0) somewhere in the middle. Then, mark positive numbers to the right of zero and negative numbers to the left.
Step 2: Locate the Critical Value
The critical value in our solution is -9. This is the point where the solution starts. Find -9 on your number line and mark it.
Step 3: Use the Correct Circle
Since our inequality is x ≥ -9 (greater than or equal to), we use a closed circle (or a filled-in circle) at -9. A closed circle indicates that -9 is included in the solution set.
If the inequality were x > -9 (greater than), we would use an open circle (or an empty circle) to show that -9 is not included in the solution set.
Step 4: Shade the Correct Direction
The inequality x ≥ -9 means that x can be any value greater than or equal to -9. On the number line, values greater than -9 are to the right of -9. So, we shade the portion of the number line to the right of -9.
The Graph
Your graph should look like this:
- A number line with 0 marked.
- A closed circle at -9.
- The line shaded to the right of -9.
This graph visually represents that any value of x that is -9 or greater is a solution to the inequality (1/3)(12-4x) ≤ 16.
Real-World Applications of Inequalities
Inequalities aren't just abstract mathematical concepts; they have practical applications in many real-world scenarios. Here are a few examples:
- Budgeting: Suppose you have a budget of $100 for groceries. If x represents the amount you spend, you can express this situation as x ≤ 100. This inequality helps you stay within your budget.
- Speed Limits: Speed limits on roads are expressed as inequalities. For example, a speed limit of 65 mph can be written as s ≤ 65, where s is your speed. This ensures safety on the road.
- Age Restrictions: Many activities, like driving or purchasing alcohol, have age restrictions. If the legal drinking age is 21, this can be expressed as a ≥ 21, where a is your age.
- Manufacturing: In manufacturing, tolerances are often expressed as inequalities. For example, the diameter of a bolt might need to be within a certain range, such as 10 mm ≤ d ≤ 10.2 mm, where d is the diameter of the bolt.
- Health and Fitness: Inequalities can be used to represent healthy ranges for various health metrics. For instance, a healthy blood pressure reading might be represented as 90 ≤ systolic ≤ 120.
Common Mistakes to Avoid
When working with inequalities, it's easy to make mistakes if you're not careful. Here are some common pitfalls to watch out for:
Forgetting to Flip the Inequality Sign
The most common mistake is forgetting to flip the inequality sign when multiplying or dividing both sides by a negative number. Remember, this step is crucial for maintaining the correct relationship between the expressions.
Misinterpreting the Inequality Symbols
Make sure you understand the difference between the inequality symbols. Less than (<) and greater than (>) do not include the endpoint, while less than or equal to (≤) and greater than or equal to (≥) do.
Graphing Errors
When graphing, double-check whether you need an open or closed circle and make sure you shade in the correct direction. Shading in the wrong direction will lead to an incorrect representation of the solution set.
Not Simplifying Properly
Before solving, always simplify the inequality as much as possible. Distribute, combine like terms, and clear fractions if necessary. A simpler inequality is easier to solve and less prone to errors.
Not Checking the Solution
After solving, it's a good idea to check your solution by plugging a value from your solution set back into the original inequality. This helps ensure that your solution is correct.
Conclusion
Graphing the solution set for the inequality (1/3)(12-4x) ≤ 16 involves solving the inequality and then representing the solution on a number line. We found that the solution is x ≥ -9, which is graphed with a closed circle at -9 and shading to the right. Inequalities are powerful tools with numerous real-world applications, from budgeting to setting speed limits. By understanding the rules and common pitfalls, you can confidently solve and graph inequalities.
For further exploration and practice with inequalities, visit Khan Academy's Inequalities Section for comprehensive lessons and exercises.
