Other than a no solution​ set, use interval notation to express the solution set and then graph the solution set on a number line.
StartFraction x Over 3 EndFraction minus five sixths less than or equals StartFraction x Over 4 EndFraction plus 1

To solve the inequality:

\[
\frac{x}{3} - \frac{5}{6} \leq \frac{x}{4} + 1
\]

### Step 1: Eliminate the fractions

We first eliminate the fractions by finding the least common denominator (LCD) of 3, 6, and 4, which is 12. We multiply every term in the inequality by 12:

\[
12 \cdot \left( \frac{x}{3} \right) - 12 \cdot \left( \frac{5}{6} \right) \leq 12 \cdot \left( \frac{x}{4} \right) + 12 \cdot 1
\]

This simplifies to:

\[
4x - 10 \leq 3x + 12
\]

### Step 2: Move variables to one side

Now, we subtract \(3x\) from both sides to get all \(x\)-terms on the left-hand side:

\[
4x - 3x - 10 \leq 12
\]

This simplifies to:

\[
x - 10 \leq 12
\]

### Step 3: Solve for \(x\)

Add 10 to both sides:

\[
x \leq 22
\]

### Step 4: Express the solution in interval notation

The solution is all values of \(x\) that are less than or equal to 22. In interval notation, this is:

\[
(-\infty, 22]
\]

### Step 5: Graph the solution on a number line

- Draw a number line.
- Place a closed dot at \(22\) to indicate that \(22\) is included (since \(x \leq 22\)).
- Shade the region to the left of 22, extending to negative infinity.

Would you like to see the graph or any further details?

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Here are 5 related questions to explore:

1. How do you handle inequalities when dividing by a negative number?
2. What are the different types of inequality symbols, and how do they affect the solution?
3. How do you solve compound inequalities?
4. How does interval notation differ for strict versus non-strict inequalities?
5. Can inequalities have no solutions? If so, when?

**Tip**: Always remember to reverse the inequality symbol when multiplying or dividing both sides by a negative number!

Other than a no solution set, use interval notation to express the solution set and then graph the solution set on a number line. StartFraction x Over 3 EndFraction minus five sixths less than or equals StartFraction x Over 4 EndFraction plus 1

Learn how to solve the linear inequality x/3 - 5/6 ≤ x/4 + 1, express the solution in interval notation, and graph it on a number line. This problem covers key algebraic concepts like inequalities and interval notation, suitable for students in Grades 8-10.

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