Solve Linear Relationships: Finding x When 8x-14 < 5x+4

Q: Why can't x be negative if the math gives x < 6?

Great question! While the inequality mathematically allows negative values, the context doesn't. Daniel can't have negative sweets! Always consider what the variable represents in real life.

Q: How do I know which person has fewer sweets from the word problem?

Look for key phrases like 'fewer than' or 'less than'. The problem states 'Iván has fewer sweets than Mariano,' so Iván's amount

Q: What if I set up the inequality backwards?

If you write $ 5x + 4 instead, you'll get \( x > 6 $. Always double-check: does your answer make sense with the original constraint?

Q: Can x equal exactly 6 in this problem?

No! When x = 6, both Iván and Mariano have the same number of sweets (32 each). But the problem says Iván has fewer sweets, so x must be strictly less than 6.

Q: Why is the answer 0 < x < 6 instead of just x < 6?

Because context matters! The variable x represents the number of sweets Daniel has. Since you can't have negative sweets, we need x > 0. Combined with x

Inequality Solutions with Contextual Constraints

Daniel has a number of sweets.

Mariano has 5 times plus 4 more sweets than Daniel

Iván has 8 times plus 14 fewer sweets than Daniel.

Iván has fewer sweets than Mariano.

What is the possible number of sweets that Daniel has in terms of x?

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Step-by-step video solution

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Step-by-step written solution

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Understand the problem

Step-by-step solution

To solve this problem, we'll contemplate the mathematical relationships between the sweets.

Step 1: Expression Setup
Mariano's sweets can be expressed as $5x + 4$ , and Iván's sweets are $8x - 14$ .
Step 2: Inequality Setup
Given that Iván has fewer sweets, establish an inequality: $8x - 14 < 5x + 4$ .
Step 3: Solve the Inequality
Simplify the inequality:
$8x - 14 < 5x + 4$
Subtract $5x$ from both sides:
$3x - 14 < 4$
Add 14 to both sides:
$3x < 18$
Divide by 3:
$x < 6$
Step 4: Considering Logical Constraints
Since sweets represent physical goods, $x$ must be positive. Therefore, $x > 0$ .
Step 5: Conclusion
Combining these inequalities, $0 < x < 6$ .

Therefore, the possible number of sweets that Daniel has is $0 < x < 6$ .

Final Answer

$0 < x < 6$

Key Points to Remember

Essential concepts to master this topic

Setup: Translate word problem into mathematical inequality expression
Technique: Solve 8x - 14 < 5x + 4 by collecting like terms
Check: Apply real-world constraint x > 0 for final answer ✓

Common Mistakes

Avoid these frequent errors

Ignoring the physical constraint that x must be positive
Don't solve 8x - 14 < 5x + 4 and stop at x < 6! This ignores that Daniel can't have negative sweets. Always consider real-world constraints: since x represents sweets, x > 0, giving final answer 0 < x < 6.

Practice Quiz

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Solve the following inequality:

$3x+4 \leq 10$

FAQ

Everything you need to know about this question

Why can't x be negative if the math gives x < 6?

Great question! While the inequality mathematically allows negative values, the context doesn't. Daniel can't have negative sweets! Always consider what the variable represents in real life.

How do I know which person has fewer sweets from the word problem?

Look for key phrases like 'fewer than' or 'less than'. The problem states 'Iván has fewer sweets than Mariano,' so Iván's amount < Mariano's amount.

What if I set up the inequality backwards?

If you write $5x + 4 < 8x - 14$ instead, you'll get $x > 6$ . Always double-check: does your answer make sense with the original constraint?

Can x equal exactly 6 in this problem?

No! When x = 6, both Iván and Mariano have the same number of sweets (32 each). But the problem says Iván has fewer sweets, so x must be strictly less than 6.

Why is the answer 0 < x < 6 instead of just x < 6?

Because context matters! The variable x represents the number of sweets Daniel has. Since you can't have negative sweets, we need x > 0. Combined with x < 6, we get 0 < x < 6.

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Solve Linear Relationships: Finding x When 8x-14 < 5x+4

Inequality Solutions with Contextual Constraints

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why can't x be negative if the math gives x < 6?

How do I know which person has fewer sweets from the word problem?

What if I set up the inequality backwards?

Can x equal exactly 6 in this problem?

Why is the answer 0 < x < 6 instead of just x < 6?

🌟 Unlock Your Math Potential

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