https://r2.erweima.ai/mathbot/image/1727222950906ruwgygm6.jpg

Let's go through the four mathematical expressions shown in the image one by one:

### 1. **Inequality:**
\[
\frac{|4x - 3|}{4} \geq 3
\]
This is an absolute value inequality. To solve this:

- First, multiply both sides of the inequality by 4:
  \[
  |4x - 3| \geq 12
  \]
  
- The absolute value inequality splits into two cases:
  - Case 1: \( 4x - 3 \geq 12 \)
  - Case 2: \( 4x - 3 \leq -12 \)

- For Case 1:
  \[
  4x \geq 15 \quad \Rightarrow \quad x \geq \frac{15}{4} = 3.75
  \]

- For Case 2:
  \[
  4x \leq -9 \quad \Rightarrow \quad x \leq \frac{-9}{4} = -2.25
  \]

Thus, the solution is:
\[
x \leq -2.25 \quad \text{or} \quad x \geq 3.75
\]

### 2. **Equation:**
\[
2x^{1/2} + x^{1/4} - 6 = 0
\]
This equation involves fractional exponents. To solve it:

- Let \( u = x^{1/4} \), so \( u^2 = x^{1/2} \). The equation becomes:
  \[
  2u^2 + u - 6 = 0
  \]

- This is a quadratic equation. Using the quadratic formula:
  \[
  u = \frac{-1 \pm \sqrt{1^2 - 4(2)(-6)}}{2(2)} = \frac{-1 \pm \sqrt{1 + 48}}{4} = \frac{-1 \pm \sqrt{49}}{4} = \frac{-1 \pm 7}{4}
  \]

- So, \( u = \frac{6}{4} = 1.5 \) or \( u = \frac{-8}{4} = -2 \).

- Since \( u = x^{1/4} \), we raise both sides to the power of 4 to solve for \( x \):
  - If \( u = 1.5 \), then \( x = (1.5)^4 = 5.0625 \).
  - If \( u = -2 \), then \( x = (-2)^4 = 16 \).

Thus, the solutions are \( x = 5.0625 \) and \( x = 16 \).

### 3. **Linear Function:**
\[
y = \frac{2x + 4}{4}
\]
This can be simplified:
\[
y = \frac{2}{4}x + \frac{4}{4} = \frac{1}{2}x + 1
\]
This is the equation of a straight line with slope \( \frac{1}{2} \) and y-intercept 1.

### 4. **Conic Section Equation:**
\[
x^2 + y^2 - 2x + 6y - 6 = 0
\]
This represents the equation of a circle. To find its standard form, we complete the square:

- For \( x^2 - 2x \), complete the square by adding and subtracting \( 1 \):
  \[
  (x - 1)^2 - 1
  \]

- For \( y^2 + 6y \), complete the square by adding and subtracting \( 9 \):
  \[
  (y + 3)^2 - 9
  \]

Substituting these into the equation:
\[
(x - 1)^2 - 1 + (y + 3)^2 - 9 - 6 = 0
\]

Simplifying:
\[
(x - 1)^2 + (y + 3)^2 = 16
\]
This is the equation of a circle with center \( (1, -3) \) and radius \( 4 \).

---

Would you like more details on any of these solutions?

Here are 5 related questions to explore further:
1. How do you solve absolute value inequalities in general?
2. What are the general steps to solve a quadratic equation by substitution?
3. How can you graph linear functions effectively?
4. What is the geometric interpretation of completing the square in conic sections?
5. How do you find the radius of a circle from its equation?

**Tip:** When solving equations with fractional exponents, it’s often useful to make substitutions to simplify the process.

Solve the following mathematical problems:
1. Solve the inequality: |4x - 3| / 4 >= 3.
2. Solve the equation: 2x^(1/2) + x^(1/4) - 6 = 0.
3. Find the simplified form of the function: y = (2x + 4) / 4.
4. Find the standard form of the conic section: x^2 + y^2 - 2x + 6y - 6 = 0.

Learn how to solve an absolute value inequality, a radical equation, simplify a linear function, and rewrite a conic section equation in standard form. This problem involves core concepts like inequalities, quadratic equations, and completing the square, suitable for high school students in grades 10-12.

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