Question

what is the solution set of 2x - 3 = 5 and 5x + 3 > -12?

Answer 1

To solve the given equations and inequalities, we'll tackle them one at a time.

1. **Solving the equation:**
\(2x - 3 = 5\)

Add 3 to both sides:
\[
2x = 5 + 3
\]
\[
2x = 8
\]

Now, divide both sides by 2:
\[
x = \frac{8}{2} = 4
\]

So, the solution to the equation is \(x = 4\).

2. **Solving the inequality:**
\(5x + 3 > -12\)

Subtract 3 from both sides:
\[
5x > -12 - 3
\]
\[
5x > -15
\]

Now, divide both sides by 5:
\[
x > -3
\]

3. **Combining the results:**
We have the solution from the equation \(x = 4\) and the solution from the inequality \(x > -3\).

Since \(4 > -3\), \(x = 4\) is included in the solution set of the inequality.

Thus, the solution set combining both parts is:
- The solution set from the equation is \( \{ 4 \} \).
- The solution set from the inequality is \( (-3, \infty) \).

Therefore, the complete solution set that satisfies both parts is:
\[
\{ 4 \} \text{ (satisfying the equation)} \text{ and } x > -3.
\]

In interval notation, the overall solution includes \(4\) within the context of the inequality:
\[
(-3, \infty) \text{ which includes } 4.
\]

So, the final result can be summarized as \(4 \in (-3, \infty)\).