Download C. Use Mental Math to Solve and Equation

Algebra 1
1.4 Write Equations and Inequalities
Vocabulary
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Equation – a mathematical sentence formed by placing the symbol = between two expressions
Inequality – a mathematical sentence formed by placing one of the symbols <, >, ≤, or ≥ between two expressions
Open Sentence – an equation or inequality that contains an algebraic expression
Solution to an Equation –a number that makes the sentence true
Solution to an Inequality – a number or set of numbers that makes the sentence true
Symbol
Meaning
Associated Words
=
is equal to
The same as
<
is less than
Fewer than
>
is greater than
More than
≤
is less than or equal to
At most; no more than
≥
is greater than or equal to
At least; no less than
The BIG Difference
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Equations: Have ONLY ONE Solution!
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Inequalities: Have MANY Solutions!!!!!
A. Translating Verbal Phrases into Equations or Inequalities
Verbal Sentence
Equation or Inequality
a. The difference of twice a number k and 8 is 12.
2k – 8 = 12
b. The product of 6 and a number n is at least 24.
6n ≥ 24
GUIDED PRACTICE:
a. A number y is no less than 5 and no more than 13.
5 ≤ y ≤ 13
b. Write an equation or an inequality: The quotient of a number p and 12 is at least 30.
p/12 ≥ 30
B. Checking to see if it is a solution
Check whether 3 is a solution of the equation or inequality.
Equation/Inequality
Substitute
Conclusion
a. 8 – 2x = 2
8 – 2(3) ? 2
2 = 2 , so 3 is a solution.
b. 4x – 5 = 6
4(3) – 5 ? 6
7 = 6 , so 3 is not a solution.
c. 2z + 5 > 12
2(3) + 5 ? 12
11 > 12, so 3 is not a solution.
d. 5 + 3n ≤ 20
5 + 3(3) ? 20
14 ≤ 20 , 3 is a solution.
GUIDED PRACTICE:
Check to see whether or not 5 is a solution of the equation or inequality.
Equation/Inequality
Substitute
Conclusion
a. 9 – x = 4
9–5?4
4 = 4, so 5 is a solution
b. b + 5 < 15
5 + 5 ? 15
10 < 15, so 5 is a solution
c. 2n + 3 ≥ 21
2(5) + 3 ? 21
13 ≥ 21, so 5 is not a solution
C. Use Mental Math to Solve and Equation
Equation
Think
Solution
Check
a. x + 4 = 10
What number plus 4 equals 10?
6
6 + 4 = 10
b. 20 – y = 8
20 minutes what number equals 8?
12
20 – 12 = 8
c.6n = 42
6 times what number equals 42?
7
6(7) = 42
d. a/5 = 9
What number divided by 5 equals 9?
45
45/5 = 9
GUIDED PRACTICE:
Solve the equation using mental math.
Equation
Think
Solution
Check
a. m + 6 = 11
What number plus 6 equals 11?
5
5 + 6 = 11
b. 5x = 40
5 times what number equals 40?
8
5(8) = 40
c. r/4 = 10
What number divided by 4 equals 10
40
40/4 = 10
d. Is 2 a solution to 4z – 5 < 3?
NOT
e.
24
f
4
6
D. Solving a Multi-Step Problem
EXAMPLE #1: The last time you and 3 friends went to a mountain bike park, you had a coupon for $10 off and paid $17 for 4 tickets.
A. What is the regular price of 4 tickets?
B. If you pay the regular price this time and share it equally, how much does each person pay?
Step 1: Write a verbal model.
Let p be the regular price of 4 tickets. Write an equation.
Regular Price – Coupon = Amount Paid
P – 10 = 17
Step 2: Use mental math to solve the equation p – 10 = 17.
Think: 10 less than what number is 17? Because 27 – 10 = 17, the solution is 27.
A. Answer: The regular price for 4 tickets is $27.
Step 3: Find Cost Per Person
$27 / 4 people = 6.75
B. Answer: $6.75 per person.
WHAT IF….Suppose that the price of 4 tickets with a half-off coupon is $15. What is each person’s share if you pay full price?
STEP 1: Write a verbal model. Let p be the regular price of 4 tickets. Write an equation.
Regular Price – Coupon = Amount Paid
r – 15 = 15
STEP 2: Use mental math to solve the equation p – 15=15.
Think: 15 less than what number is 15? Because 30 – 15 = 15, the solution is 30. So the full price is $30.
STEP 3: Find the Cost Per Person
$30/4 = 7.5
Answer: $7.50 per person
EXAMPLE #2: A basketball player scored 351 points last year. If the player plays 18 games this year, will an average of 20 points per
game be enough to beat last year’s total?
STEP 1: Write a verbal model. Let p be the average number of points per game. Write an inequality.
Number of Games • Number of Points Per Game = Total Points Last Year
18 • p > 351
STEP 2: Check that 20 is a solution of the in equality18p > 351.
Because 18(20) = 360 and 360 > 351, 20 is a solution
Answer: An average of 20 points per game will be enough.
WHAT IF… Suppose that the player plays 16 games. Would an average of 22 points per game be enough to beat last year’s total?
STEP 1: Write a verbal model. Let p be the average number of points per game. Write an inequality.
Number of Games • Number of Points Per Game = Total Points Last Year
STEP 2: Check that 22 is a solution of the in equality16p > 351. Because 16(22) = 352 and 352 > 351, 22 is a solution.