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resuelve inecuacion de 1er grado e indica el mayor valor entero de x

Let's solve the inequality given in the image:

\[
\frac{x + 3}{4} - \frac{x - 4}{5} > \frac{x - 1}{2} - \frac{x + 2}{3}
\]

**Step 1: Find a common denominator.**

For the left-hand side, the common denominator between 4 and 5 is 20.

\[
\frac{x + 3}{4} = \frac{5(x + 3)}{20} = \frac{5x + 15}{20}, \quad \frac{x - 4}{5} = \frac{4(x - 4)}{20} = \frac{4x - 16}{20}
\]

So, the left-hand side becomes:

\[
\frac{5x + 15}{20} - \frac{4x - 16}{20} = \frac{5x + 15 - 4x + 16}{20} = \frac{x + 31}{20}
\]

For the right-hand side, the common denominator between 2 and 3 is 6.

\[
\frac{x - 1}{2} = \frac{3(x - 1)}{6} = \frac{3x - 3}{6}, \quad \frac{x + 2}{3} = \frac{2(x + 2)}{6} = \frac{2x + 4}{6}
\]

So, the right-hand side becomes:

\[
\frac{3x - 3}{6} - \frac{2x + 4}{6} = \frac{3x - 3 - 2x - 4}{6} = \frac{x - 7}{6}
\]

**Step 2: Rewrite the inequality.**

Now, the inequality is:

\[
\frac{x + 31}{20} > \frac{x - 7}{6}
\]

**Step 3: Cross multiply to eliminate the fractions.**

Cross multiplying gives:

\[
6(x + 31) > 20(x - 7)
\]

**Step 4: Expand both sides.**

\[
6x + 186 > 20x - 140
\]

**Step 5: Move all terms involving \(x\) to one side and constant terms to the other side.**

\[
186 + 140 > 20x - 6x
\]

\[
326 > 14x
\]

**Step 6: Solve for \(x\).**

\[
x < \frac{326}{14} \approx 23.2857
\]

Since the problem asks for the largest integer value of \(x\), the largest integer less than 23.2857 is **23**.

Thus, the largest integer value of \(x\) that satisfies the inequality is \( \boxed{23} \).

Would you like any further details or have any questions?

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Here are some related questions that might interest you:

1. How would the solution change if the inequality were flipped?
2. What are the steps to solve inequalities with different denominators?
3. Can you provide an example of a more complex inequality?
4. How does cross multiplication work in the context of inequalities?
5. How do we check the solution of an inequality?
6. What happens if the inequality involves quadratic terms?
7. Can this inequality be solved using a graphical method?
8. How does solving an inequality differ from solving an equation?

**Tip:** When solving inequalities, always be cautious about the direction of the inequality sign, especially when multiplying or dividing by negative numbers.

Learn how to solve a first-degree inequality involving fractions step-by-step. Find the largest integer value of x that satisfies the inequality. Suitable for Grades 9-12.

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