A. Translate the following English phrases and sentences to symbolic form. 1. 8 less twice a number
2. the product of 3 and the square of the sum of two numbers
3. one-fourth of the difference of a number and 5
4. the square of a number increased by twice of itself
5. the quotient of the sum and the product of three numbers
6. The sum of the squares of two numbers is less than or equal to the square of
their sum. 7. If x is an element of the intersection of sets A and B, then x is an element of
both sets A and B. 8. If A is a subset of B and B is not a subset of A, then the union of A and B is B. 9. The product of two integers is also an integer. 10. There exist no two real numbers such that their difference is equal to their
sum. B. Translate the following mathematical sentences to English form. 11. 4￾￾￾
￾ t ￾
￾
12. ￾
￾ ￾ h
￾ ￾ ￾ ￾ h
￾
13. ￾￾ ⺂ ￾￾￾h ￾ ￾￾￾h t ￾
14. ￾ ￾ ￾ ￾ ￾ ￾ ￾ ￾ ￾ ￾ ￾￾ 15. ￾ ￾ ￾ ￾ ล￾ ￾ ￾ t ￾ ￾ ￾ ￾ ￾ t ￾￾

### A. Translate the following English phrases and sentences to symbolic form.

1. **8 less twice a number**  
   Symbolic form: \( 2x - 8 \)

2. **The product of 3 and the square of the sum of two numbers**  
   Symbolic form: \( 3(x + y)^2 \)

3. **One-fourth of the difference of a number and 5**  
   Symbolic form: \( \frac{1}{4}(x - 5) \)

4. **The square of a number increased by twice of itself**  
   Symbolic form: \( x^2 + 2x \)

5. **The quotient of the sum and the product of three numbers**  
   Symbolic form: \( \frac{x + y + z}{xyz} \)

6. **The sum of the squares of two numbers is less than or equal to the square of their sum.**  
   Symbolic form: \( x^2 + y^2 \leq (x + y)^2 \)

7. **If \( x \) is an element of the intersection of sets \( A \) and \( B \), then \( x \) is an element of both sets \( A \) and \( B \).**  
   Symbolic form: \( x \in A \cap B \Rightarrow x \in A \) and \( x \in B \)

8. **If \( A \) is a subset of \( B \) and \( B \) is not a subset of \( A \), then the union of \( A \) and \( B \) is \( B \).**  
   Symbolic form: \( A \subseteq B \) and \( B \nsubseteq A \Rightarrow A \cup B = B \)

9. **The product of two integers is also an integer.**  
   Symbolic form: \( \forall a, b \in \mathbb{Z}, \, ab \in \mathbb{Z} \)

10. **There exist no two real numbers such that their difference is equal to their sum.**  
    Symbolic form: \( \nexists x, y \in \mathbb{R} \, \text{such that} \, x - y = x + y \)

### B. Translate the following mathematical sentences to English form.

11. **\( 4t \)**  
    Translation: "Four times \( t \)."

12. **\( h^2 \)**  
    Translation: "The square of \( h \)."

13. **\( h \times t \)**  
    Translation: "The product of \( h \) and \( t \)."

14. **\( t \leq 5 \)**  
    Translation: "The variable \( t \) is less than or equal to 5."

15. **\( t + 7 \geq 10 \)**  
    Translation: "The sum of \( t \) and 7 is greater than or equal to 10."

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Would you like more details or have any questions? Here are five related questions:

1. How do you translate a compound inequality into symbolic form?
2. What are some common phrases in English that correspond to mathematical operations?
3. How would you translate "the sum of a number and its square" into symbolic form?
4. What is the symbolic form for "twice the square of a number decreased by 3"?
5. How do you express "a number is at most 10" in symbolic form?

**Tip:** When translating phrases to symbolic form, pay close attention to key words like "less," "more," "product," "sum," and "difference" to accurately reflect the intended operations.

A. Translate the following English phrases and sentences to symbolic form. 1. 8 less twice a number 2. the product of 3 and the square of the sum of two numbers 3. one-fourth of the difference of a number and 5 4. the square of a number increased by twice of itself 5. the quotient of the sum and the product of three numbers 6. The sum of the squares of two numbers is less than or equal to the square of their sum. 7. If x is an element of the intersection of sets A and B, then x is an element of both sets A and B. 8. If A is a subset of B and B is not a subset of A, then the union of A and B is B. 9. The product of two integers is also an integer. 10. There exist no two real numbers such that their difference is equal to their sum. B. Translate the following mathematical sentences to English form. 11. 4t 12. h^2 13. h × t 14. t ≤ 5 15. t + 7 ≥ 10

Learn how to translate complex English phrases into symbolic mathematical form and vice versa. This detailed guide covers key algebraic concepts, set theory, and inequalities, including step-by-step solutions for Grades 9-12 students.

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