To solve the given equations and inequalities, we'll tackle them one at a time.
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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 \(x = 4\).
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Solving the inequality:
\(5x + 3 > -12\)Subtract 3 from both sides: \[ 5x > -12 - 3 \] \[ 5x > -15 \]
Now, divide both sides by 5: \[ x > -3 \]
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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 \(4 \in (-3, \infty)\).