Inequality Solutions: Finding When -3x + 15 < 3x < 4x + 8 Holds True

Q: Why do I need to split this into two separate inequalities?

Compound inequalities like \( a actually mean two conditions must be true at the same time: \( a AND \( b . Solving them separately makes the algebra much clearer!

Q: How do I find the final answer when I have two separate solutions?

Look for the intersection (overlap) of your two solutions. In this case, $ x > -8 $ AND $ x > 2.5 $ gives us $ x > 2.5 $ since it's the more restrictive condition.

Q: What if my two inequalities don't overlap?

If there's no overlap, then the original compound inequality has no solution! This happens when the conditions contradict each other, like $ x > 5 $ AND \( x .

Q: How can I check my final answer?

Pick a test value that satisfies your answer (like x = 3 for $ x > 2.5 $) and substitute it into the original compound inequality. All parts should be true!

Q: Do I always take the more restrictive condition?

Yes! When you have two conditions like $ x > a $ and $ x > b $, the intersection is $ x > \max(a,b) $. The larger value is more restrictive and represents the overlap.

Compound Inequalities with Variable Separation

Find when the inequality is satisfied:

$-3x+15<3x<4x+8$

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Understand the problem

Find when the inequality is satisfied:

$-3x+15<3x<4x+8$

Step-by-step solution

To solve this problem effectively, we will proceed by solving each inequality separately:

Step 1: Solve the inequality $-3x + 15 < 3x$ .

Add $3x$ to both sides to get: $15 < 6x$ .
Divide both sides by $6$ to solve for $x$ : $x > \frac{15}{6}$ , simplifying to $x > 2.5$ .

Step 2: Solve the second inequality $3x < 4x + 8$ .

Subtract $3x$ from both sides to isolate $x$ : $0 < x + 8$ .
Subtract $8$ from both sides to find: $-8 < x$ .

Step 3: Find the overlap of the solutions.

The solution to the inequalities $-8 < x$ and $x > 2.5$ is simply $x > 2.5$ since $x > 2.5$ is more restrictive.

Thus, the values of $x$ that satisfy the original compound inequality are those for which $x > 2.5$ .

Therefore, the solution to the problem is $2.5 < x$ .

Final Answer

$2.5 < x$

Key Points to Remember

Essential concepts to master this topic

Rule: Split compound inequalities into separate inequalities first
Technique: For -3x + 15 < 3x, add 3x to both sides: 15 < 6x
Check: Test x = 3 in original: -9 + 15 < 9 < 20 → 6 < 9 < 20 ✓

Common Mistakes

Avoid these frequent errors

Trying to solve the entire compound inequality at once
Don't attempt to solve -3x + 15 < 3x < 4x + 8 as one step = confusing algebra and wrong answers! The variables appear in different positions, making direct manipulation impossible. Always split into two separate inequalities: -3x + 15 < 3x AND 3x < 4x + 8, then find their intersection.

Practice Quiz

Test your knowledge with interactive questions

Solve the following inequality:

$3x+4 \leq 10$

FAQ

Everything you need to know about this question

Why do I need to split this into two separate inequalities?

Compound inequalities like $a < b < c$ actually mean two conditions must be true at the same time: $a < b$ AND $b < c$ . Solving them separately makes the algebra much clearer!

How do I find the final answer when I have two separate solutions?

Look for the intersection (overlap) of your two solutions. In this case, $x > -8$ AND $x > 2.5$ gives us $x > 2.5$ since it's the more restrictive condition.

What if my two inequalities don't overlap?

If there's no overlap, then the original compound inequality has no solution! This happens when the conditions contradict each other, like $x > 5$ AND $x < 2$ .

How can I check my final answer?

Pick a test value that satisfies your answer (like x = 3 for $x > 2.5$ ) and substitute it into the original compound inequality. All parts should be true!

Do I always take the more restrictive condition?

Yes! When you have two conditions like $x > a$ and $x > b$ , the intersection is $x > \max(a,b)$ . The larger value is more restrictive and represents the overlap.

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