1. 1.4 – Writing Equations and Inequalities

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

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