If the set of x values satisfying    (3x+10)/2 >2  and  3x-6<x-2  is (A,B) ,  what is A+B?

We are given two inequalities:

\[
\frac{3x + 10}{2} > 2 \quad \text{and} \quad 3x - 6 < x - 2
\]

Let's solve them step by step.

### First inequality:
\[
\frac{3x + 10}{2} > 2
\]
Multiply both sides by 2 to eliminate the fraction:
\[
3x + 10 > 4
\]
Now subtract 10 from both sides:
\[
3x > -6
\]
Divide by 3:
\[
x > -2
\]

### Second inequality:
\[
3x - 6 < x - 2
\]
First, subtract \(x\) from both sides:
\[
2x - 6 < -2
\]
Now add 6 to both sides:
\[
2x < 4
\]
Finally, divide by 2:
\[
x < 2
\]

### Combine the two results:
From the first inequality, we have \(x > -2\), and from the second inequality, we have \(x < 2\).

Thus, the solution set is:
\[
-2 < x < 2
\]

This corresponds to the interval \((-2, 2)\). Therefore, \(A = -2\) and \(B = 2\).

### To find \(A + B\):
\[
A + B = -2 + 2 = 0
\]

The value of \(A + B\) is \( \boxed{0} \).

Would you like further details, or do you have any questions?

Here are 5 related questions to explore:

1. What if one inequality was non-strict, like \( \frac{3x + 10}{2} \geq 2 \)?
2. How does changing a coefficient in an inequality impact the solution range?
3. What happens if we add an additional condition, like \(x \geq 0\)?
4. How can interval notation help in representing solutions graphically?
5. How do absolute value inequalities differ from linear inequalities?

**Tip:** Always check if your inequality flips when dividing by negative numbers!

If the set of x values satisfying (3x+10)/2 > 2 and 3x-6 < x-2 is (A,B), what is A+B?

Learn how to solve the compound inequality (3x + 10)/2 > 2 and 3x - 6 < x - 2 with a step-by-step solution. This problem demonstrates key algebraic concepts such as inequality solving, combining solution sets, and interval notation. Suitable for students in Grades 8-10.

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