Question

Solve the inequality. Graph and check your solutions.
three fourths plus a less than or equals minus five fourths
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Part 1
a
▼
less than
greater than
greater than or equals
less than or equals
  
enter your response here
​(Simplify your​ answer.)

Answer 1

To solve the inequality \(\frac{3}{4} + a \leq -\frac{5}{4}\), we first want to isolate \(a\).

1. **Subtract \(\frac{3}{4}\) from both sides:**

\[
a \leq -\frac{5}{4} - \frac{3}{4}
\]

2. **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.

- **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.

- **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: \(a \leq -2\).