To solve the inequality \(\frac{3}{4} + a \leq -\frac{5}{4}\), we first want to isolate \(a\).
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Subtract \(\frac{3}{4}\) from both sides:
\[ a \leq -\frac{5}{4} - \frac{3}{4} \]
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Combine the fractions on the right side:
\[ a \leq -\left(\frac{5}{4} + \frac{3}{4}\right) = -\frac{5 + 3}{4} = -\frac{8}{4} = -2 \]
Thus, we have:
\[ a \leq -2 \]
Graphing the Solution
To graph the solution \(a \leq -2\):
- Draw a number line.
- Mark the point \(-2\).
- Shade to the left of \(-2\) to indicate all values less than \(-2\).
- Use a closed circle at \(-2\) to symbolize that \(-2\) is included in the solution (because of the "less than or equal to").
Solution Summary
The solution to the inequality is:
\[ a \leq -2 \]
This means \(a\) can take any value that is less than or equal to \(-2\).
Checking the Solution
To check the solution, we can pick a value within and outside the solution set and substitute it back into the original inequality.
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Check \(a = -3\) (a value less than \(-2\)):
\(\frac{3}{4} + (-3) = \frac{3}{4} - \frac{12}{4} = -\frac{9}{4}\)
\(-\frac{9}{4} \leq -\frac{5}{4}\) is true.
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Check \(a = -1\) (a value greater than \(-2\)):
\(\frac{3}{4} + (-1) = \frac{3}{4} - \frac{4}{4} = -\frac{1}{4}\)
\(-\frac{1}{4} \leq -\frac{5}{4}\) is false.
The checks confirm that our solution is correct: \(a \leq -2\).