https://file.aiquickdraw.com/mathbot/file/1727444108409pn4pgekt.jpg

The inequality presented in the image is:

\[
\frac{x + 1}{3} < \frac{7}{5} + 2
\]

Let's solve the inequality step by step.

### Step 1: Simplify the right-hand side

First, we simplify the expression on the right-hand side of the inequality:

\[
\frac{7}{5} + 2 = \frac{7}{5} + \frac{10}{5} = \frac{17}{5}
\]

So the inequality becomes:

\[
\frac{x + 1}{3} < \frac{17}{5}
\]

### Step 2: Eliminate the denominators by multiplying by 15

To eliminate the fractions, multiply both sides of the inequality by 15 (which is the least common multiple of 3 and 5):

\[
15 \times \frac{x + 1}{3} < 15 \times \frac{17}{5}
\]

This simplifies to:

\[
5(x + 1) < 3 \times 17
\]

\[
5(x + 1) < 51
\]

### Step 3: Distribute and solve for \( x \)

Now distribute the 5 on the left-hand side:

\[
5x + 5 < 51
\]

Subtract 5 from both sides:

\[
5x < 46
\]

Divide by 5:

\[
x < \frac{46}{5} = 9.2
\]

### Step 4: Interpret the solution

The inequality \( x < 9.2 \) means that the value of \( x \) must be less than 9.2. Now, looking at the multiple-choice options (12, 15, 16, 20), none of these values are less than 9.2.

It seems none of the provided options are correct, but based on the solution, the answer should be any number less than 9.2.

Would you like further details or assistance on this? Here are some related questions for practice:

1. What steps would change if the inequality sign was \( \geq \) instead of \( < \)?
2. How would you solve this if there was a negative sign in front of \( x \)?
3. What if the fractions had different denominators, like 4 and 7, instead of 3 and 5?
4. How does multiplying or dividing by negative numbers affect the inequality?
5. How would you check your solution after solving an inequality?

**Tip**: When dealing with inequalities, always be cautious when multiplying or dividing by negative numbers; the inequality sign flips!

Which value is a solution to the inequality: (x + 1) / 3 < 7 / 5 + 2?

Step-by-step guide on solving the algebraic inequality (x + 1) / 3 < 7 / 5 + 2 by simplifying fractions, eliminating denominators, and solving for x. Ideal for students in Grades 8-10, this problem covers key algebraic concepts including inequalities and fractions.

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