Help me solve this inequality, x^2-3/4≥6/4

To solve the inequality \( x^2 - \frac{3}{4} \geq \frac{6}{4} \), follow these steps:

### Step 1: Simplify the inequality
First, simplify the constants on the right side:
\[
\frac{6}{4} = \frac{3}{2}
\]
The inequality becomes:
\[
x^2 - \frac{3}{4} \geq \frac{3}{2}
\]

### Step 2: Isolate the \( x^2 \) term
Add \( \frac{3}{4} \) to both sides to isolate \( x^2 \):
\[
x^2 \geq \frac{3}{2} + \frac{3}{4}
\]

### Step 3: Combine the fractions on the right-hand side
To add the fractions \( \frac{3}{2} \) and \( \frac{3}{4} \), find a common denominator:
\[
\frac{3}{2} = \frac{6}{4}
\]
\[
\frac{6}{4} + \frac{3}{4} = \frac{9}{4}
\]
So the inequality now is:
\[
x^2 \geq \frac{9}{4}
\]

### Step 4: Solve the inequality for \( x \)
To solve for \( x \), take the square root of both sides. Remember to consider both the positive and negative roots:
\[
x \geq \frac{3}{2} \quad \text{or} \quad x \leq -\frac{3}{2}
\]

### Final Answer:
The solution to the inequality is:
\[
x \leq -\frac{3}{2} \quad \text{or} \quad x \geq \frac{3}{2}
\]

This means \( x \) must be less than or equal to \( -\frac{3}{2} \) or greater than or equal to \( \frac{3}{2} \).

Would you like to explore this topic further, or do you have any other questions?

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Here are 5 related questions you might find interesting:
1. How do you solve quadratic inequalities using the sign chart method?
2. What are the general steps for solving polynomial inequalities?
3. How does one graph the solution to a quadratic inequality?
4. What is the difference between solving a quadratic equation and a quadratic inequality?
5. How do absolute value inequalities differ from quadratic inequalities?

**Tip:** Always remember to check the boundary points in inequalities, especially when dealing with quadratic expressions!

Learn how to solve the inequality x^2 - 3/4 ≥ 6/4 with a detailed step-by-step solution. This problem covers key algebraic concepts related to quadratic inequalities. Suitable for students in Grades 9-12.

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