Hey guys! Let's break down this inequality step by step. Inequalities might seem intimidating at first, but trust me, they're totally manageable once you get the hang of it. We're going to simplify both sides, isolate the variable, and find the range of values that make the inequality true. So, grab your thinking caps and let’s dive in!

Understanding the Basics

Before we jump into solving, let's quickly recap what an inequality is. Unlike an equation, which states that two expressions are equal, an inequality indicates that two expressions are not necessarily equal. Common inequality symbols include:

• :> Greater than
• :< Less than
• :≥ Greater than or equal to
• :≤ Less than or equal to

Our goal is to find all values of x that satisfy the given inequality: 3 + 4 + 6x + 10 * 2 < 27 + 64.

Step-by-Step Solution

1. Simplify Both Sides

The first thing we need to do is simplify both sides of the inequality by combining like terms. Let's start with the left side:

3 + 4 + 6x + 10 * 2

Following the order of operations (PEMDAS/BODMAS), we first perform the multiplication:

10 * 2 = 20

Now, we add the constants:

3 + 4 + 20 = 27

So, the left side simplifies to:

27 + 6x

Next, let's simplify the right side of the inequality:

27 + 64

Adding these numbers gives us:

27 + 64 = 91

Now our inequality looks much simpler:

27 + 6x < 91

2. Isolate the Variable Term

Our next goal is to isolate the term with x on one side of the inequality. To do this, we need to get rid of the constant term on the left side. We can subtract 27 from both sides of the inequality:

27 + 6x - 27 < 91 - 27

This simplifies to:

6x < 64

3. Solve for x

Now, we need to solve for x by dividing both sides of the inequality by 6:

6x / 6 < 64 / 6

This gives us:

x < 64/6

We can simplify the fraction 64/6 by dividing both the numerator and the denominator by 2:

64/6 = 32/3

So, our solution is:

x < 32/3

4. Convert to Mixed Number (Optional)

Sometimes, it's helpful to express the solution as a mixed number. To convert 32/3 to a mixed number, we divide 32 by 3:

32 ÷ 3 = 10 with a remainder of 2

So, 32/3 is equal to 10 2/3. Therefore, our solution can also be written as:

x < 10 2/3

Final Answer

The solution to the inequality 3 + 4 + 6x + 10 * 2 < 27 + 64 is:

x < 32/3 or x < 10 2/3

This means that any value of x that is less than 32/3 (or 10 2/3) will satisfy the original inequality. To verify, you can pick a value less than 10 2/3 and plug it back into the original inequality to see if it holds true.

Visualizing the Solution

Understanding the solution becomes even clearer when visualized on a number line. Imagine a number line stretching from negative infinity to positive infinity. Our solution, x < 32/3, represents all the numbers to the left of 32/3 (or 10 2/3).

To represent this on a number line:

1. Draw a number line: Mark zero and some key points around 10 2/3.
2. Locate 10 2/3: Place a mark at approximately 10 2/3 on the number line.
3. Use an open circle: Since our inequality is strictly less than (<), we use an open circle at 10 2/3 to indicate that 10 2/3 itself is not included in the solution.
4. Shade the region to the left: Shade everything to the left of the open circle. This shaded region represents all the values of x that satisfy x < 10 2/3.

This visual representation provides an intuitive understanding of the solution set, making it easier to grasp the concept.

Common Mistakes to Avoid

When solving inequalities, it's easy to make a few common mistakes. Here are some pitfalls to watch out for:

1. Forgetting to Flip the Inequality Sign

One of the most critical rules in solving inequalities is that when you multiply or divide both sides by a negative number, you must flip the direction of the inequality sign. For example, if you have -2x < 6, dividing by -2 requires you to change the inequality to x > -3.

2. Incorrect Order of Operations

Always follow the correct order of operations (PEMDAS/BODMAS). Perform operations inside parentheses first, then exponents, then multiplication and division (from left to right), and finally addition and subtraction (from left to right). Mixing up the order can lead to incorrect simplifications.

3. Arithmetic Errors

Simple arithmetic errors can easily throw off your solution. Double-check your calculations, especially when dealing with fractions or negative numbers.

4. Not Distributing Properly

When distributing a number across parentheses, make sure to multiply every term inside the parentheses. For example, 3(x + 2) should be expanded to 3x + 6, not just 3x + 2.

5. Misunderstanding the Solution Set

Be clear about what the solution represents. For example, x > 5 means all numbers greater than 5, not including 5 itself. An open circle on a number line indicates that the endpoint is not included, while a closed circle indicates that it is.

By being mindful of these common mistakes, you can improve your accuracy and confidence when solving inequalities.

Real-World Applications

Inequalities aren't just abstract mathematical concepts; they have numerous real-world applications. Understanding inequalities can help you make informed decisions in various aspects of life.

1. Budgeting and Finance

When managing a budget, you often deal with inequalities. For example, you might want to ensure that your expenses are less than or equal to your income. This can be represented as:

Expenses ≤ Income

Similarly, when investing, you might set a minimum return on investment:

Return on Investment ≥ Minimum Acceptable Return

2. Health and Fitness

In health and fitness, inequalities can help define healthy ranges. For instance, a healthy blood pressure reading might be defined as:

Systolic Blood Pressure < 120 and Diastolic Blood Pressure < 80

Or, you might set a goal to exercise for at least 30 minutes a day:

Exercise Time ≥ 30 minutes

3. Engineering and Construction

Engineers use inequalities to ensure structures are safe and meet certain specifications. For example, the maximum load a bridge can support must be greater than the expected load:

Maximum Load Capacity > Expected Load

4. Business and Economics

Businesses use inequalities to analyze costs, revenues, and profits. For example, a company might need to ensure that its revenue is greater than its costs to make a profit:

Revenue > Costs

5. Everyday Decision Making

Even in everyday situations, we use inequalities without realizing it. For example, when deciding whether to take an umbrella, you might think:

Chance of Rain > Threshold for Taking Umbrella

Understanding inequalities allows you to quantify and analyze these situations, leading to better decisions.

Practice Problems

To solidify your understanding of solving inequalities, let's work through a few practice problems.

Practice Problem 1

Solve the inequality: 5x - 3 > 12

Solution:

1. Add 3 to both sides: 5x > 15
2. Divide by 5: x > 3

So, the solution is x > 3.

Practice Problem 2

Solve the inequality: -(2x + 1) ≤ 7

Solution:

1. Distribute the negative sign: -2x - 1 ≤ 7
2. Add 1 to both sides: -2x ≤ 8
3. Divide by -2 (and flip the inequality sign): x ≥ -4

So, the solution is x ≥ -4.

Practice Problem 3

Solve the inequality: 3(x - 2) < 5x + 4

Solution:

1. Distribute the 3: 3x - 6 < 5x + 4
2. Subtract 3x from both sides: -6 < 2x + 4
3. Subtract 4 from both sides: -10 < 2x
4. Divide by 2: -5 < x

So, the solution is x > -5.

By working through these practice problems, you can reinforce your skills and gain confidence in solving inequalities.

Conclusion

So, there you have it! Solving inequalities might seem tricky at first, but with a bit of practice, you'll become a pro in no time. Remember to simplify, isolate the variable, and watch out for those negative signs! You've got this! Keep practicing, and you'll be solving inequalities like a math whiz in no time!