Let's break down and solve this inequality step by step! Inequalities might seem tricky at first, but with a bit of arithmetic and algebra, you'll see they're totally manageable. We're going to simplify both sides of the inequality first, then isolate 'x' to find our solution.

Step 1: Simplify Both Sides

First, let's simplify the left-hand side (LHS) of the inequality:

3 + 4 + 6x + 10 * 2

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

10 * 2 = 20

Now, substitute this back into the LHS:

3 + 4 + 6x + 20

Combine the constants:

3 + 4 + 20 = 27

So, the simplified LHS is:

27 + 6x

Now, let's simplify the right-hand side (RHS) of the inequality:

27 + 64 = 91

So, the simplified inequality is:

27 + 6x < 91

Step 2: Isolate the Variable

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

27 + 6x - 27 < 91 - 27

This simplifies to:

6x < 64

Now, to completely isolate 'x', we need to divide both sides by the coefficient of 'x', which is 6:

6x / 6 < 64 / 6

This simplifies to:

x < 64/6

We can further simplify the fraction 64/6 by dividing both the numerator and the denominator by their greatest common divisor, which is 2:

x < 32/3

Step 3: Express the Solution

So, the solution to the inequality is:

x < 32/3

This means that any value of 'x' that is less than 32/3 will satisfy the original inequality. To express this as a mixed number:

32 ÷ 3 = 10 with a remainder of 2

So,

32/3 = 10 2/3

Therefore, the solution can also be written as:

x < 10 2/3

In decimal form, this is approximately:

x < 10.67

So, any number less than approximately 10.67 will satisfy the inequality.

Step 4: Verification

To verify our solution, we can pick a value for 'x' that is less than 32/3 (or 10.67) and plug it back into the original inequality to see if it holds true. Let's pick x = 10:

3 + 4 + 6(10) + 10 * 2 < 27 + 64

Simplify the LHS:

3 + 4 + 60 + 20 < 27 + 64

87 < 91

Since 87 is indeed less than 91, our solution is correct!

Expressing the Solution Graphically

We can represent the solution x < 32/3 on a number line. Here's how:

1. Draw a number line: Draw a straight line and mark some numbers on it, including 0, 10, and 11. Make sure to include 32/3 (or 10 2/3) on your number line.
2. Place an open circle at 32/3: Since the inequality is x < 32/3 (and not x ≤ 32/3), we use an open circle to indicate that 32/3 is not included in the solution.
3. Shade to the left of 32/3: Shade the region of the number line to the left of 32/3. This shaded region represents all the values of x that are less than 32/3.
4. Draw an arrow: At the left end of the shaded region, draw an arrow to indicate that the solution extends infinitely in the negative direction.

This graphical representation visually shows all the possible values of x that satisfy the inequality.

Common Mistakes to Avoid

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

• Forgetting to Distribute: If you have a term multiplied by an expression in parentheses, make sure to distribute it correctly. For example, if you have 2(x + 3), it becomes 2x + 6, not 2x + 3.
• Incorrectly Combining Like Terms: Be careful when combining like terms. Make sure you're only combining terms that have the same variable and exponent. For example, you can combine 3x and 5x to get 8x, but you can't combine 3x and 5x^2.
• Not Flipping the Inequality Sign: When you multiply or divide both sides of an inequality by a negative number, you need to flip the inequality sign. For example, if you have -2x < 6, dividing by -2 gives x > -3 (note the flipped sign).
• Order of Operations Errors: Always follow the order of operations (PEMDAS/BODMAS) when simplifying expressions. Parentheses/Brackets first, then Exponents/Orders, then Multiplication and Division (from left to right), and finally Addition and Subtraction (from left to right).
• Verification: Not verifying the solution by plugging it back into the original equation to check if it holds true

By being aware of these common mistakes, you can avoid them and solve inequalities more accurately.

Real-World Applications

Inequalities aren't just abstract math concepts; they show up in many real-world situations. Here are a few examples:

• Budgeting: When you're creating a budget, you might use inequalities to represent how much money you can spend on different categories. For example, if you want to spend no more than $100 on groceries, you could write the inequality g ≤ 100, where g represents the amount you spend on groceries.
• Setting Limits: Inequalities are often used to set limits or restrictions. For example, a roller coaster might have a height restriction, such as h ≥ 48 inches, where h represents a person's height. This inequality means that you must be at least 48 inches tall to ride the roller coaster.
• Determining Profit: Businesses use inequalities to determine the number of products they need to sell to make a profit. For example, if the cost to produce a product is $5 and the selling price is $10, the profit per product is $5. If the business has fixed costs of $1000, they need to sell enough products to cover those costs. The inequality could be written as 5x ≥ 1000, where x represents the number of products sold.
• Grading: In education, inequalities can be used to determine the range of scores needed to achieve a certain grade. For example, to get an A in a class, you might need a score of at least 90%. This could be represented by the inequality s ≥ 90, where s represents your score.
• Speed Limits: Speed limits on roads are a common example of inequalities. If the speed limit is 65 mph, it means you should drive at a speed s ≤ 65, where s is your speed.

Understanding inequalities can help you make informed decisions and solve problems in various aspects of life.

Practice Problems

Now that you understand the basics of solving inequalities, here are a few practice problems to test your skills:

1. Solve: 2x + 5 < 11
2. Solve: -3x - 7 ≥ 5
3. Solve: 4(x - 2) > 8
4. Solve: 1/2x + 3 ≤ 7
5. Solve: 6 - x < 10

Try solving these problems on your own, and then check your answers with the solutions below:

1. x < 3
2. x ≤ -4
3. x > 4
4. x ≤ 8
5. x > -4

If you got all the problems correct, congratulations! You have a good understanding of how to solve inequalities. If you missed any problems, review the steps and try again.

Inequalities are fundamental in math and have numerous applications in real life. By understanding and practicing how to solve them, you enhance your problem-solving skills and analytical thinking. Keep practicing, and you'll master inequalities in no time!