Solving For 'h': Inequality And Equation Exploration

Hey math enthusiasts! Let's dive into a fun problem. We're given an equation: (3x + h)/5 - (2x - 5h)/4 = 1. The catch? This equation must be true when x is less than or equal to 7 (x ≤ 7). Our mission, should we choose to accept it, is to find the smallest possible value of h. Sounds like a blast, right? This problem blends algebra with a touch of inequality, and we'll break it down step-by-step to make sure everything clicks. We'll start by tackling the equation itself, then we'll see how the x ≤ 7 condition impacts our solution and ultimately helps us pinpoint that elusive smallest value of h. Ready? Let's get started!

Unraveling the Equation: The Initial Steps

Alright, guys, first things first: let's get rid of those pesky fractions! The easiest way to do this is to multiply both sides of the equation by the least common multiple (LCM) of the denominators, which are 5 and 4. The LCM of 5 and 4 is 20. So, we'll multiply the entire equation by 20. Doing this gives us:

• 20 * [(3x + h)/5 - (2x - 5h)/4] = 20 * 1

Distributing the 20, we get:

• 20 * (3x + h)/5 - 20 * (2x - 5h)/4 = 20

Now, let's simplify:

• 4 * (3x + h) - 5 * (2x - 5h) = 20

Expanding the terms:

• 12x + 4h - 10x + 25h = 20

Combining like terms:

• 2x + 29h = 20

We're making good progress! Now we have a simplified version of our equation. It is important to keep in mind, we still have to figure out the value of h that works with the inequality x ≤ 7. This will play a crucial role later on in finding the smallest possible value for h.

Isolating x and Understanding the Constraint

Our next move is to isolate x in the equation. This will give us a better understanding of how x and h relate to each other. From our simplified equation (2x + 29h = 20), let's subtract 29h from both sides:

• 2x = 20 - 29h

Now, divide both sides by 2 to solve for x:

• x = (20 - 29h) / 2

Great! We've successfully isolated x. But remember, the problem states that x must be less than or equal to 7 (x ≤ 7). This is where the inequality comes into play. Let's substitute our expression for x into the inequality:

• (20 - 29h) / 2 ≤ 7

Now we need to solve this inequality for h. We're getting closer to that smallest h value! To start, multiply both sides by 2:

• 20 - 29h ≤ 14

Subtract 20 from both sides:

• -29h ≤ -6

Finally, divide both sides by -29. Important Note: When you divide or multiply both sides of an inequality by a negative number, you must flip the direction of the inequality sign. So, we get:

• h ≥ 6/29

So, after all the calculations, we have found that h must be greater than or equal to 6/29 (h ≥ 6/29). This is the key to determining the smallest value of h. Now we are cooking with gas!

Pinpointing the Smallest Value of h

Okay, team, we've done the heavy lifting! We've simplified the equation, incorporated the inequality, and solved for h. Now comes the easy part: finding the smallest value of h. Our final inequality states that h must be greater than or equal to 6/29 (h ≥ 6/29). Think about what this means. Any value of h that is equal to or bigger than 6/29 will satisfy the original equation given the condition that x ≤ 7. So, what is the smallest possible value? That's right, it's 6/29!

Why? Because 6/29 is the lower bound. Any value smaller than 6/29 would violate the inequality we derived. Therefore, the smallest value of h that satisfies the given conditions is exactly 6/29. It's a beautiful thing when all the pieces of the puzzle fit together perfectly, isn't it? We successfully navigated the equation, used the inequality to guide us, and found the smallest possible value of h. Give yourselves a pat on the back, you all! You totally nailed it.

Let's wrap up with a summary of the important points:

• Equation simplification: We cleared fractions and combined like terms to simplify the initial equation.
• Isolating x: We solved for x in terms of h, which provided a link between x and h.
• Incorporating the Inequality: We replaced x in the inequality x ≤ 7 with our solved expression, enabling us to get an inequality in terms of h.
• Solving for h: We manipulated the inequality to solve for h, revealing a lower bound.
• Identifying the smallest h: The smallest h value is the lower bound we derived from our inequality, 6/29.

Conclusion: The Final Answer

We did it, guys! We started with a seemingly complex equation and inequality, but by breaking it down step-by-step, we successfully found the smallest possible value of h that satisfies the conditions. The answer, my friends, is h = 6/29. Math problems like this are great because they challenge our problem-solving skills and help us to think critically. Remember, the key is to stay organized, take it one step at a time, and never be afraid to ask for help or clarify the steps along the way. Congrats on making it through this problem! Keep up the amazing work.

And that's a wrap. We have uncovered the smallest value of h! The final answer is h = 6/29.

Keep practicing, keep learning, and keep the math adventures going!