Question

Solve the inequality. Graph and check your solutions.
five thirds
greater thannplus3
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Part 1
n
▼
less than or equals
less than
greater than
greater than or equals
  
enter your response here
​(Type an integer or a simplified​ fraction.)

Answer 1

To solve the inequality given in the prompt:

\[
\frac{5}{3} > n + 3
\]

We first rearrange the inequality to solve for \( n \).

1. Subtract 3 from both sides:

\[
\frac{5}{3} - 3 > n
\]

2. To combine the terms on the left, we need to express 3 as a fraction with a denominator of 3:

\[
3 = \frac{9}{3}
\]

Thus, we have:

\[
\frac{5}{3} - \frac{9}{3} > n
\]

3. Subtracting the fractions:

\[
\frac{5 - 9}{3} > n
\]
\[
\frac{-4}{3} > n
\]

This can be rewritten as:

\[
n < -\frac{4}{3}
\]

So, the solution to the inequality is:

\[
n < -\frac{4}{3}
\]

### Graphing the solution:

You can graph this on a number line:

- Draw a number line.
- Mark the point \(-\frac{4}{3}\) (which is approximately -1.33).
- Place an open circle at \(-\frac{4}{3}\) to indicate that this value is not included in the solution.
- Shade the region to the left of \(-\frac{4}{3}\) to represent all values less than \(-\frac{4}{3}\).

### Checking the solution:

To check the solution, let's take a value less than \(-\frac{4}{3}\). For example, let’s test \( n = -2 \):

1. Substitute \( n = -2 \) into the original inequality:

\[
\frac{5}{3} > -2 + 3
\]
\[
\frac{5}{3} > 1
\]

Since \( \frac{5}{3} \) (which is approximately 1.67) is indeed greater than 1, our solution is verified.

Now, let’s check a value greater than \(-\frac{4}{3}\), for example \( n = -1 \):

\[
\frac{5}{3} > -1 + 3
\]
\[
\frac{5}{3} > 2
\]

Since \( \frac{5}{3} \) is not greater than 2, this confirms that values greater than \(-\frac{4}{3}\) do not satisfy the inequality.

Final answer:

\( n < -\frac{4}{3} \)