To find the solution set for the equations \(3x - 2 = 7\) and the inequality \(4x + 5 \leq 21\), we need to solve each one separately.
Step 1: Solve the equation \(3x - 2 = 7\)
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Add 2 to both sides: \[ 3x = 7 + 2 \] \[ 3x = 9 \]
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Divide both sides by 3: \[ x = \frac{9}{3} \] \[ x = 3 \]
Step 2: Solve the inequality \(4x + 5 \leq 21\)
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Subtract 5 from both sides: \[ 4x \leq 21 - 5 \] \[ 4x \leq 16 \]
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Divide both sides by 4: \[ x \leq \frac{16}{4} \] \[ x \leq 4 \]
Step 3: Combine the results
The solution to the equation gives us \(x = 3\). The solution to the inequality indicates that \(x\) can be any value less than or equal to 4.
Since 3 is less than or equal to 4, it satisfies the inequality.
Final Solution Set
Thus, the complete solution set is:
- The equation \(3x - 2 = 7\) gives \(x = 3\).
- The inequality \(4x + 5 \leq 21\) gives \(x \leq 4\).
Conclusion: The solution set that satisfies both the equation and the inequality is: \[ x = 3 \] And since it is also valid for the inequality, we state the final solution:
\[ { x | x = 3 \text{ and } x \leq 4 } \] or simply, \[ { 3 } \] Which is a subset of \( (-\infty, 4] \).