As a final review of the algebraic expressions we have studied, we will learn how to find the value of an expression by substitution and how to express relationships between quantities using equations and inequalities. This content will always be needed when you use algebraic expressions.
Substitution and the Value of an Expression
Replacing a letter with a specific number is called substitution. The result calculated after substitution is the value of the expression.
Tip: When substituting, replace the letter with a number in parentheses to prevent sign errors.
- When x = 3, 2x + 1 ＝ 2×(3)+1 ＝ 7
- When x = −2, 3x − 5 ＝ 3×(−2)−5 ＝ −6 − 5 ＝ −11
- When a = −3, a² ＝ (−3)² ＝ 9 (not −3²!)
When x = −4 and y = 3, find the values of the following expressions.
(1) 2x − y (2) x² + 3y
Answer:
(1) 2×(−4) − (3) ＝ −8 − 3 ＝ −11
(2) (−4)² + 3×(3) ＝ 16 + 9 ＝ 25
Equations and Inequalities
Use ＝ (equals sign) to express that two quantities are equal, and use the inequality signs ＜, ＞, ≦, ≧ to express their relative sizes.
- Equation: An expression connected by an equals sign. Example: 2x + 3 ＝ 7
- Inequality: An expression connected by an inequality sign. Example: 3a − 1 ≧ 5
Meanings of inequality signs:
a ＜ b: a is less than b
a ＞ b: a is greater than b
a ≦ b: a is less than or equal to b (including a＝b)
a ≧ b: a is greater than or equal to b (including a＝b)
Changing Words into Equations and Inequalities
When changing a relationship between quantities into an algebraic expression, first determine whether the quantities are “equal,” “greater,” or “less.”
- “Buying 5 items costing a yen each costs b yen” → 5a ＝ b
- “A height of x cm is taller than 150cm” → x ＞ 150
- “After spending 800 yen from x yen, at least 200 yen remained” → x − 800 ≧ 200
- “Distributing a candies to 6 people, b candies each, leaves 3 candies” → a ＝ 6b + 3
Express each relationship between quantities as an equation or inequality.
(1) Four notebooks costing x yen each and y pencils costing 80 yen each were bought for a total of 1000 yen.
(2) The number obtained by adding 5 to three times a number a is less than 20.
Answer:
(1) 4x + 80y ＝ 1000
(2) 3a + 5 ＜ 20 (“less than” does not include equality, so use ＜)
- Use parentheses for substitution! In particular, watch the difference between (−3)² and −3² when substituting a negative number
- “Equal” → ＝, “relative size” → ＜, ＞, ≦, ≧
- “At least/at most” includes equality (≧, ≦) / “greater than/less than” does not (＞, ＜)
- For word problems, first decide what quantities are equal or which is greater, then create the expression
Practice Problems
- When a = −5, find the value of a² − 2a + 3.
- Write an expression for the following quantity: The total distance traveled for 3 hours at x km/h and for y hours at 60 km/h is 200 km.
- Express the following relationship as an inequality: Buy 5 books costing a yen each and pay 3000 yen so that at least 100 yen remains as change.
Answers and Explanations
- Answer: 38
Explanation: Substitute a=−5: (−5)² − 2×(−5) + 3 = 25 + 10 + 3 = 38. Watch the signs in (−5)² = 25 and −2×(−5) = +10. - Answer: 3x + 60y ＝ 200
Explanation: The first distance is 3x km and the second distance is 60y km. Since their total is 200 km, the equation is 3x + 60y = 200. - Answer: 3000 − 5a ≧ 100
Explanation: The cost of 5 books is 5a yen. Change ＝ 3000 − 5a. “At least 100 yen” includes 100, so use ≧: 3000 − 5a ≧ 100.