Solving Linear Inequalities: A Step-by-Step Guide

Let's break down how to solve the inequality 7(x+1) - 6x ≥ -5. Inequalities might seem tricky at first, but with a clear, step-by-step approach, you'll be solving them like a pro in no time! We'll go through each stage, explaining the logic and the math involved, so you fully understand how to arrive at the solution. So, grab your pen and paper, and let's get started!

Understanding Linear Inequalities

Before diving into the problem, let's briefly recap what linear inequalities are all about. A linear inequality is similar to a linear equation, but instead of an equals sign (=), it has one of the following inequality symbols: < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to). Solving an inequality means finding the range of values for the variable (in our case, 'x') that make the inequality true. The key difference between solving equations and inequalities is that when you multiply or divide both sides of an inequality by a negative number, you must flip the direction of the inequality sign. Keep this in mind, as it’s a common pitfall! Also remember that the solution to an inequality is often a set of numbers, not just a single value, which is why we express the answer as a solution set. Now, let's tackle the problem at hand.

Step-by-Step Solution

Our goal is to isolate 'x' on one side of the inequality. Here's how we do it:

1. Distribute: First, we need to get rid of the parentheses. We do this by distributing the 7 across the (x + 1) term:

7 * x + 7 * 1 - 6x ≥ -5

This simplifies to:

7x + 7 - 6x ≥ -5

2. Combine Like Terms: Next, we combine the 'x' terms on the left side of the inequality:

(7x - 6x) + 7 ≥ -5

This gives us:

x + 7 ≥ -5

3. Isolate x: Now, we want to isolate 'x' by getting rid of the +7 on the left side. We do this by subtracting 7 from both sides of the inequality:

x + 7 - 7 ≥ -5 - 7

This simplifies to:

x ≥ -12

Expressing the Solution Set

The solution to the inequality is x ≥ -12. This means that any value of 'x' that is greater than or equal to -12 will satisfy the original inequality. To express this as a solution set, we use the following notation:

{x | x ≥ -12}

This is read as "the set of all x such that x is greater than or equal to -12."

Verification

It's always a good idea to check your answer to make sure it's correct. We can do this by picking a value of 'x' that satisfies our solution (x ≥ -12) and plugging it back into the original inequality. Let's pick x = -12 (the boundary value) and x = -10 (a value greater than -12).

• Check with x = -12:

  7((-12) + 1) - 6(-12) ≥ -5

  7(-11) + 72 ≥ -5

  -77 + 72 ≥ -5

  -5 ≥ -5 (This is true)

• Check with x = -10:

  7((-10) + 1) - 6(-10) ≥ -5

  7(-9) + 60 ≥ -5

  -63 + 60 ≥ -5

  -3 ≥ -5 (This is also true)

Since the inequality holds true for both values, we can be confident that our solution set is correct.

Common Mistakes to Avoid

• Forgetting to Flip the Inequality Sign: As mentioned earlier, if you multiply or divide both sides of an inequality by a negative number, you must flip the direction of the inequality sign. For example, if you have -x > 5, dividing both sides by -1 gives you x < -5 (not x > -5).
• Incorrect Distribution: Make sure to distribute correctly, paying attention to signs. For example, a common mistake is to incorrectly distribute a negative sign. For example -2(x - 3) = -2x + 6, not -2x - 6.
• Combining Unlike Terms: You can only combine terms that have the same variable and exponent. For example, you can combine 3x and 5x to get 8x, but you cannot combine 3x and 5x². Always double-check which terms are like terms before simplifying!
• Arithmetic Errors: Simple arithmetic errors can lead to incorrect answers. Double-check your calculations, especially when dealing with negative numbers.

Advanced Tips and Tricks

• Graphing the Solution Set: You can represent the solution set of an inequality on a number line. For x ≥ -12, you would draw a closed circle (or square bracket) at -12 and shade the number line to the right, indicating all values greater than -12 are included.
• Interval Notation: Another way to represent the solution set is using interval notation. For x ≥ -12, the interval notation is [-12, ∞). The square bracket indicates that -12 is included in the solution, and the parenthesis indicates that infinity is not included (since it's not a specific number).
• Compound Inequalities: Sometimes you'll encounter compound inequalities, which are two inequalities joined by "and" or "or." For example, x > 3 and x < 7. To solve compound inequalities, you need to solve each inequality separately and then combine the solutions based on the "and" or "or" condition. If it's "and", you need to find the overlap, and if it's an "or", you need to combine everything.

Real-World Applications

Inequalities aren't just abstract mathematical concepts; they're used in many real-world situations. Here are a few examples:

• Budgeting: Suppose you have a budget of $100 for groceries. If 'x' represents the amount you spend, then the inequality x ≤ 100 represents your budget constraint.
• Speed Limits: The speed limit on a highway might be 65 mph. If 'v' represents your speed, then the inequality v ≤ 65 represents the legal speed limit.
• Manufacturing: A company might need to produce at least 1000 units of a product to make a profit. If 'n' represents the number of units produced, then the inequality n ≥ 1000 represents the production requirement.
• Grading: To get an A in a class, you might need to score at least 90%. If 's' represents your score, then the inequality s ≥ 90 represents the requirement for getting an A.

Conclusion

So, to wrap things up, we've successfully solved the inequality 7(x + 1) - 6x ≥ -5 and found the solution set to be {x | x ≥ -12}. Remember the key steps: distribute, combine like terms, isolate the variable, and express the solution set correctly. Don't forget to double-check your work, especially when dealing with negative numbers or flipping inequality signs. With practice, you'll become more comfortable and confident in solving linear inequalities. Keep practicing, and you'll be an inequality master in no time! And remember, math isn't just about finding the right answer; it's about understanding the process and developing problem-solving skills that you can apply in many different areas of your life. Good luck, and happy solving!