1. 1.4 – Writing Equations and Inequalities

2. An equation is a mathematical sentence formed by placing the symbol = between two expressions.
inequality is a mathematical sentence formed by placing one of the symbols <, ≤, >, or ≥ between two expressions.

3. = < ≤ > ≥ Symbol Meaning Associated Words is equal to
the same as
fewer than
is less than
at most, no more than
is less than or equal to
is greater than
more than
is greater than or equal to
at least, no less than

4. Example 1: Write an equation or inequality from the given verbal sentence.
The difference of twice a number k and 8 is 12.
The product of 6 and a number n is at least 24.
A number y is no less than 5 and no more than 13.
The quotient of a number p and 12 is at least 30.
The sum of twice a number r and 3 is 11.
Verbal Sentence
Equation or Inequality
The difference of twice a number k and 8 is 12.
The product of 6 and a number n is at least 24.
A number y is no less than 5 and no more than 13.
The quotient of a number p and 12 is at least 30.
The sum of twice a number r and 3 is 11.
Verbal Sentence
Equation or Inequality
The difference of twice a number k and 8 is 12.
The product of 6 and a number n is at least 24.
A number y is no less than 5 and no more than 13.
The quotient of a number p and 12 is at least 30.
The sum of twice a number r and 3 is 11.
2k – 8 = 12

5. 2k – 8 = 12 2k – 8 = 12 Verbal Sentence Equation or Inequality
The difference of twice a number k and 8 is 12.
The product of 6 and a number n is at least 24.
A number y is no less than 5 and no more than 13.
The quotient of a number p and 12 is at least 30.
The sum of twice a number r and 3 is 11.
Verbal Sentence
Equation or Inequality
The difference of twice a number k and 8 is 12.
The product of 6 and a number n is at least 24.
A number y is no less than 5 and no more than 13.
The quotient of a number p and 12 is at least 30.
The sum of twice a number r and 3 is 11.
Verbal Sentence
Equation or Inequality
The difference of twice a number k and 8 is 12.
The product of 6 and a number n is at least 24.
A number y is no less than 5 and no more than 13.
The quotient of a number p and 12 is at least 30.
The sum of twice a number r and 3 is 11.
2k – 8 = 12
2k – 8 = 12

6. 6n ≥ 24 2k – 8 = 12 2k – 8 = 12 Verbal Sentence Equation or Inequality
The difference of twice a number k and 8 is 12.
The product of 6 and a number n is at least 24.
6n ≥ 24
A number y is no less than 5 and no more than 13.
The quotient of a number p and 12 is at least 30.
The sum of twice a number r and 3 is 11.
Verbal Sentence
Equation or Inequality
The difference of twice a number k and 8 is 12.
The product of 6 and a number n is at least 24.
A number y is no less than 5 and no more than 13.
The quotient of a number p and 12 is at least 30.
The sum of twice a number r and 3 is 11.
Verbal Sentence
Equation or Inequality
The difference of twice a number k and 8 is 12.
The product of 6 and a number n is at least 24.
A number y is no less than 5 and no more than 13.
The quotient of a number p and 12 is at least 30.
The sum of twice a number r and 3 is 11.
2k – 8 = 12
2k – 8 = 12

7. 6n ≥ 24 2k – 8 = 12 2k – 8 = 12 5 ≤ y ≤ 13 Verbal Sentence
Equation or Inequality
The difference of twice a number k and 8 is 12.
The product of 6 and a number n is at least 24.
6n ≥ 24
A number y is no less than 5 and no more than 13.
The quotient of a number p and 12 is at least 30.
The sum of twice a number r and 3 is 11.
Verbal Sentence
Equation or Inequality
The difference of twice a number k and 8 is 12.
The product of 6 and a number n is at least 24.
A number y is no less than 5 and no more than 13.
The quotient of a number p and 12 is at least 30.
The sum of twice a number r and 3 is 11.
Verbal Sentence
Equation or Inequality
The difference of twice a number k and 8 is 12.
The product of 6 and a number n is at least 24.
A number y is no less than 5 and no more than 13.
The quotient of a number p and 12 is at least 30.
The sum of twice a number r and 3 is 11.
2k – 8 = 12
2k – 8 = 12
5 ≤ y ≤ 13

8. 6n ≥ 24 2k – 8 = 12 2k – 8 = 12 5 ≤ y ≤ 13 Verbal Sentence
Equation or Inequality
The difference of twice a number k and 8 is 12.
The product of 6 and a number n is at least 24.
6n ≥ 24
A number y is no less than 5 and no more than 13.
The quotient of a number p and 12 is at least 30.
The sum of twice a number r and 3 is 11.
Verbal Sentence
Equation or Inequality
The difference of twice a number k and 8 is 12.
The product of 6 and a number n is at least 24.
A number y is no less than 5 and no more than 13.
The quotient of a number p and 12 is at least 30.
The sum of twice a number r and 3 is 11.
Verbal Sentence
Equation or Inequality
The difference of twice a number k and 8 is 12.
The product of 6 and a number n is at least 24.
A number y is no less than 5 and no more than 13.
The quotient of a number p and 12 is at least 30.
The sum of twice a number r and 3 is 11.
2k – 8 = 12
2k – 8 = 12
5 ≤ y ≤ 13

9. 6n ≥ 24 2r + 3 = 11 2k – 8 = 12 2k – 8 = 12 5 ≤ y ≤ 13 Verbal Sentence
Equation or Inequality
The difference of twice a number k and 8 is 12.
The product of 6 and a number n is at least 24.
6n ≥ 24
A number y is no less than 5 and no more than 13.
The quotient of a number p and 12 is at least 30.
The sum of twice a number r and 3 is 11.
2r + 3 = 11
Verbal Sentence
Equation or Inequality
The difference of twice a number k and 8 is 12.
The product of 6 and a number n is at least 24.
A number y is no less than 5 and no more than 13.
The quotient of a number p and 12 is at least 30.
The sum of twice a number r and 3 is 11.
Verbal Sentence
Equation or Inequality
The difference of twice a number k and 8 is 12.
The product of 6 and a number n is at least 24.
A number y is no less than 5 and no more than 13.
The quotient of a number p and 12 is at least 30.
The sum of twice a number r and 3 is 11.
2k – 8 = 12
2k – 8 = 12
5 ≤ y ≤ 13

10. 2k – 8 = 12 Equation or Inequality Verbal Sentence
A number q is at least 5 and less than 17
Two-thirds of a number b is at most 24.
The quotient of a number y divided into 10 is 5.
2k – 8 = 12

11. 5 ≤ q < 17 2k – 8 = 12 Equation or Inequality Verbal Sentence
A number q is at least 5 and less than 17
5 ≤ q < 17
Two-thirds of a number b is at most 24.
The quotient of a number y divided into 10 is 5.
2k – 8 = 12

12. 5 ≤ q < 17 (⅔)b ≤ 24 2k – 8 = 12 Equation or Inequality
Verbal Sentence
Equation or Inequality
A number q is at least 5 and less than 17
5 ≤ q < 17
Two-thirds of a number b is at most 24.
(⅔)b ≤ 24
The quotient of a number y divided into 10 is 5.
2k – 8 = 12

13. Verbal Sentence
Equation or Inequality
A number q is at least 5 and less than 17
5 ≤ q < 17
Two-thirds of a number b is at most 24.
(⅔)b ≤ 24
The quotient of a number y divided into 10 is 5.
(10 ÷ y) = 5
2k – 8 = 12

14. Example 2: Check whether 5 is a solution of the equation or inequality.
Substitute
Conclusion
24 – 3d = 9
3x + 4 = 18
2w – 7 ≤ 3
4 + 3p > 19

15. 24 – 3d = 9 24 – 3(5) = 9 yes 3x + 4 = 18 2w – 7 ≤ 3 4 + 3p > 19
Equation/
Inequality
Substitute
Conclusion
24 – 3d = 9
24 – 3(5) = 9
yes
3x + 4 = 18
2w – 7 ≤ 3
4 + 3p > 19

16. 24 – 3d = 9 24 – 3(5) = 9 yes 3x + 4 = 18 3(5) + 4 = 19 no 2w – 7 ≤ 3
Equation/
Inequality
Substitute
Conclusion
24 – 3d = 9
24 – 3(5) = 9
yes
3x + 4 = 18
3(5) + 4 = 19 no
2w – 7 ≤ 3
4 + 3p > 19

17. 24 – 3d = 9 24 – 3(5) = 9 yes 3x + 4 = 18 3(5) + 4 = 19 no 2w – 7 ≤ 3
Equation/
Inequality
Substitute
Conclusion
24 – 3d = 9
24 – 3(5) = 9
yes
3x + 4 = 18
3(5) + 4 = 19 no
2w – 7 ≤ 3
2(5) – 7 = 3
Is 3 ≤ 3
4 + 3p > 19

18. 24 – 3d = 9 24 – 3(5) = 9 yes 3x + 4 = 18 3(5) + 4 = 19 no 2w – 7 ≤ 3
Equation/
Inequality
Substitute
Conclusion
24 – 3d = 9
24 – 3(5) = 9
yes
3x + 4 = 18
3(5) + 4 = 19 no
2w – 7 ≤ 3
2(5) – 7 = 3
Is 3 ≤ 3
4 + 3p > 19
4 + 3(5) = 19
Is 19 > 19