To solve the system of equations and inequalities you've provided, we will address each part separately.
1. Solve the equation: \( 2x - 3 = 5 \)
Start by isolating \( x \):
\[ 2x - 3 = 5 \]
Add 3 to both sides:
\[ 2x = 5 + 3 \] \[ 2x = 8 \]
Now, divide both sides by 2:
\[ x = \frac{8}{2} = 4 \]
2. Solve the inequality: \( 5x + 3 > -12 \)
Start by isolating \( x \):
Subtract 3 from both sides:
\[ 5x > -12 - 3 \] \[ 5x > -15 \]
Now divide both sides by 5:
\[ x > -3 \]
Solution Set
Now, we have the solution from the equation \( x = 4 \) and the solution from the inequality \( x > -3 \).
Since we are looking for values of \( x \) that satisfy both the equation and the inequality, we need to check if \( 4 \) satisfies the inequality:
\[ 4 > -3 \]
This condition is true.
Final Solution Set
The solution set of the entire system is:
\[ { x \mid x = 4 \text{ and } x > -3 } \]
In interval notation, it can be expressed as:
\[ { 4 } \]
or simply \( x = 4 \) since it is a specific value that satisfies both conditions.
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