Solving Inequalities: -4w - 10 ≥ 40 Explained

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Let's break down how to solve the inequality −4w−10≥40-4w - 10 \geq 40. Inequalities might seem tricky, but they're really just equations with a twist! Instead of finding one exact answer, we're finding a range of possible values that make the statement true. So, grab your favorite beverage, and let’s dive in! We'll go through it step-by-step, so it's super clear and easy to follow.

Step-by-Step Solution

Alright, let’s get right to it. Our mission is to isolate w on one side of the inequality. This involves a few key moves, much like solving a regular equation, but with a small watch-out that we'll cover.

1. Isolate the Term with w

First, we want to get the term with w (which is −4w-4w) by itself on one side. To do this, we need to get rid of that pesky −10-10. How? By adding 1010 to both sides of the inequality. Remember, whatever we do to one side, we have to do to the other to keep things balanced!

−4w−10+10≥40+10-4w - 10 + 10 \geq 40 + 10

This simplifies to:

−4w≥50-4w \geq 50

Now, the term with w is nicely isolated. We're halfway there!

2. Solve for w

Next up, we need to get w all by itself. Right now, it’s being multiplied by −4-4. To undo that multiplication, we need to divide both sides by −4-4. Here’s where that special watch-out comes in:

Important Rule: When you multiply or divide both sides of an inequality by a negative number, you have to flip the direction of the inequality sign.

So, let's do it:

−4w−4≤50−4\frac{-4w}{-4} \leq \frac{50}{-4}

Notice that the $\geq`` sign flipped to $\leq``! This is super important.

Simplifying this gives us:

w≤−504w \leq -\frac{50}{4}

We can further simplify the fraction to:

w≤−252w \leq -\frac{25}{2}

Or, as a decimal:

w≤−12.5w \leq -12.5

3. Understanding the Solution

So, what does w≤−12.5w \leq -12.5 actually mean? It means that w can be any number that is less than or equal to −12.5-12.5. That includes numbers like −13-13, −15-15, −20-20, and so on. Basically, anything to the left of −12.5-12.5 on the number line.

Visualizing the Solution

It can be helpful to visualize this on a number line. Imagine a number line stretching from negative infinity to positive infinity. Find −12.5-12.5 on that line. Since w can be equal to −12.5-12.5, we use a closed circle (or a square bracket, depending on your preference) at −12.5-12.5. Then, we shade everything to the left of −12.5-12.5, indicating that all those values are part of the solution.

Why Flip the Inequality Sign?

Great question! The reason we flip the inequality sign when multiplying or dividing by a negative number has to do with how negative numbers work. Let’s think about a simple example:

We know that 2<42 < 4 is true.

Now, let’s multiply both sides by −1-1:

−1∗2-1 * 2 and −1∗4-1 * 4 which gives us −2-2 and −4-4.

Is −2<−4-2 < -4? Nope! −2-2 is actually greater than −4-4. So, we need to flip the sign to make the statement true: −2>−4-2 > -4.

This principle applies to all inequalities. Multiplying or dividing by a negative number essentially reverses the order of the numbers on the number line, so we need to flip the inequality sign to maintain the truth of the statement.

Common Mistakes to Avoid

When solving inequalities, there are a couple of common pitfalls you'll want to steer clear of:

  1. Forgetting to Flip the Sign: As we've emphasized, this is the big one! Always remember to flip the inequality sign when multiplying or dividing by a negative number. It's super easy to forget in the heat of the moment, so make it a conscious step in your process. Maybe even write yourself a little reminder note until it becomes second nature!

  2. Incorrectly Applying Operations: Just like with equations, it's crucial to apply operations correctly and to both sides of the inequality. Double-check your arithmetic, especially when dealing with negative numbers.

  3. Misinterpreting the Solution: Make sure you understand what your solution actually means. For example, if you get w>5w > 5, that means w can be any number greater than 5, not less than or equal to 5. It can be helpful to test a number that should work, and a number that shouldn't work, to confirm you understand your solution.

  4. Not simplifying completely: Always simplify the solution to its simplest form, like reducing the fraction -50/4 to -25/2 or -12.5. This makes the answer clear and easily understandable.

Real-World Applications

Inequalities aren't just abstract math concepts; they pop up in all sorts of real-world situations. Think about scenarios where there's a minimum or maximum limit, a budget constraint, or a range of acceptable values.

For example:

  • Budgeting: You have a budget of $50\$50 for groceries. If you've already spent $20\$20, the inequality 20+x≤5020 + x \leq 50 represents how much more you can spend (x).
  • Speed Limits: The speed limit on a highway is 65 mph. Your speed (s) must satisfy the inequality s≤65s \leq 65.
  • Height Requirements: To ride a roller coaster, you must be at least 48 inches tall. Your height (h) must satisfy the inequality h≥48h \geq 48.

Understanding and solving inequalities helps you make informed decisions in these types of situations.

Practice Problems

Want to put your new skills to the test? Try solving these inequalities:

  1. 3x+5<143x + 5 < 14
  2. −2y−8≥6-2y - 8 \geq 6
  3. 5a+10≤2a−25a + 10 \leq 2a - 2

Solutions:

  1. x<3x < 3
  2. y≤−7y \leq -7
  3. a≤−4a \leq -4

Conclusion

So, there you have it! Solving the inequality −4w−10≥40-4w - 10 \geq 40 isn't so daunting after all. Just remember to isolate the variable, flip the sign when multiplying or dividing by a negative number, and understand what your solution means. With a little practice, you'll be solving inequalities like a pro! Keep practicing, and don't be afraid to ask for help when you need it. Happy solving, guys! You got this! And always remember the golden rule of flipping the sign when you multiply or divide by a negative number. Good luck!