To solve an inequality, the approach depends on the type of inequality and the function involved. Here's a general guide on how to solve different types of inequalities:

1. Linear Inequality

Example: $2x + 3 > 7$

Steps:

1. Isolate the variable on one side: $2x > 7 - 3$ $2x > 4$
2. Divide by the coefficient of the variable: $x > \frac{4}{2}$ $x > 2$

The solution is $x > 2$.

2. Quadratic Inequality

Example: $x^2 - 4x - 5 \leq 0$

Steps:

1. Factor the quadratic expression (if possible): $(x - 5)(x + 1) \leq 0$
2. Find the roots (set each factor equal to zero): $x - 5 = 0 \quad \text{or} \quad x + 1 = 0$ $x = 5 \quad \text{or} \quad x = -1$
3. Determine the intervals around the roots:
   • Test points in the intervals $(-\infty, -1)$, $(-1, 5)$, and $(5, \infty)$.
   • The solution to the inequality is where the expression is less than or equal to zero.

Final Answer: The solution is $-1 \leq x \leq 5$.

3. Absolute Value Inequality

Example: $|2x - 3| < 7$

Steps:

1. Write the compound inequality: $-7 < 2x - 3 < 7$
2. Isolate the variable in the middle: $-7 + 3 < 2x < 7 + 3$ $-4 < 2x < 10$
3. Divide by the coefficient of $x$: $-2 < x < 5$

Final Answer: The solution is $-2 < x < 5$.

4. Rational Inequality

Example: $\frac{x + 2}{x - 3} \geq 1$

Steps:

1. Rewrite the inequality to have 0 on one side: $\frac{x + 2}{x - 3} - 1 \geq 0$ $\frac{x + 2 - (x - 3)}{x - 3} \geq 0$ $\frac{5}{x - 3} \geq 0$
2. Solve for the critical points:
   • The inequality is undefined at $x = 3$.
   • Analyze the sign of $\frac{5}{x - 3}$.

Final Answer: The solution depends on the sign analysis; usually, this is determined by testing values on intervals divided by critical points.

Would you like a more detailed explanation or a specific type of inequality solved?

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Here are five related questions:

1. How do you solve a system of inequalities?
2. Can you explain the concept of absolute value in more detail?
3. How do you graph a linear inequality on a coordinate plane?
4. What are the steps to solving a quadratic inequality using a sign chart?
5. How do rational inequalities differ from polynomial inequalities?

Tip: When solving inequalities, always remember to flip the inequality sign if you multiply or divide by a negative number.