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### Matched Exercise 1.7
Rewrite the following statements as intervals:
1. \( x \leq 5 \)
2. The function \( g(x) = x^2 \) has no zeros.
3. The square of a number is greater than or equal to the number itself.

Let's address each statement:

1. **\( x \leq 5 \) as an interval:**
   \[
   (-\infty, 5]
   \]

2. **The function \( g(x) = x^2 \) has no zeros:**
   This is a fact-based statement about the function \( g(x) = x^2 \). Since \( x^2 = 0 \) when \( x = 0 \), it does have a zero. Thus, this statement is false.

3. **The square of a number is greater than or equal to the number itself:**
   Let \( x \) be any real number. Then the statement can be expressed as:
   \[
   x^2 \geq x
   \]
   Rearranging this inequality, we get:
   \[
   x^2 - x \geq 0 \implies x(x - 1) \geq 0
   \]
   The solution to this inequality can be expressed as:
   \[
   (-\infty, 0] \cup [1, \infty)
   \]

Would you like any more details or have any questions?

Here are 5 related questions that expand on the previous information:

1. How do you express the inequality \( x \geq -3 \) as an interval?
2. Can you explain why the function \( f(x) = x^2 \) has no negative values?
3. How do you solve the inequality \( x^2 < x \) and express it as an interval?
4. What are the zeros of the function \( h(x) = x^3 - x \)?
5. How can you verify if a given point is a zero of a polynomial function?

**Tip:** When solving inequalities, always consider the critical points where the expression equals zero, and test intervals around these points to determine where the inequality holds true.

Rewrite the following statements as intervals:
1. x ≤ 5
2. The function g(x) = x^2 has no zeros.
3. The square of a number is greater than or equal to the number itself.

This problem asks to rewrite mathematical statements into intervals and analyze quadratic functions like x^2 = 0. Key concepts include inequalities, quadratic functions, and interval notation. Suitable for students in Grades 9-12.

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