Solve for x.
2 x left parenthesis x minus 3 right parenthesis greater than or equals 0


To solve the inequality \(2x(x - 3) \geq 0\), follow these steps:

### Step 1: Factor the expression
The inequality is already factored:
\[
2x(x - 3) \geq 0
\]
Since \(2\) is a positive constant, it does not affect the inequality. So, focus on the factors \(x\) and \((x - 3)\).

### Step 2: Determine the critical points
The critical points are found by setting each factor equal to zero:
\[
x = 0 \quad \text{and} \quad x - 3 = 0 \implies x = 3
\]

### Step 3: Test the intervals
The critical points divide the number line into three intervals:

1. \(x < 0\)
2. \(0 \leq x < 3\)
3. \(x \geq 3\)

Let's test a point from each interval to determine the sign of the expression in that interval:

- **For \(x < 0\):** Choose \(x = -1\):
  \[
  2(-1)(-1 - 3) = 2(-1)(-4) = 8 \quad (\text{positive})
  \]

- **For \(0 \leq x < 3\):** Choose \(x = 1\):
  \[
  2(1)(1 - 3) = 2(1)(-2) = -4 \quad (\text{negative})
  \]

- **For \(x \geq 3\):** Choose \(x = 4\):
  \[
  2(4)(4 - 3) = 2(4)(1) = 8 \quad (\text{positive})
  \]

### Step 4: Write the solution
The expression is non-negative (greater than or equal to zero) in the intervals where it is positive or zero.

- From the tests above, the inequality \(2x(x - 3) \geq 0\) is satisfied when \(x \leq 0\) or \(x \geq 3\).

Thus, the solution is:
\[
x \leq 0 \quad \text{or} \quad x \geq 3
\]

This can also be written as:
\[
(-\infty, 0] \cup [3, \infty)
\]

Would you like more details, or have any questions?

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**Relative Questions:**

1. How do you solve inequalities with more complex factorizations?
2. What is the effect of multiplying an inequality by a negative number?
3. Can you explain how to solve quadratic inequalities graphically?
4. How do you solve absolute value inequalities?
5. How does the solution of inequalities change when dealing with rational expressions?

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

Learn how to solve the quadratic inequality 2x(x - 3) ≥ 0 through factoring and critical point analysis. This solution covers algebraic concepts related to inequalities and quadratic expressions. Suitable for students in Grades 9-11.

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