algebra i 1.4 write equations and inequalities. vocab equation – a mathematical sentence formed by...
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VOCABEquation – 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
VOCABOpen Sentence – an equation or inequality that contains an algebraic expressionSolution to an Equation –a number that makes the sentence trueSolution to an Inequality – a number or set of numbers that makes the sentence true
1.4 Write Equations and 1.4 Write Equations and InequalitiesInequalities
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
1.4 Write Equations and 1.4 Write Equations and InequalitiesInequalities
The The BIGBIG Difference Difference
Equations:Equations:Have ONLY ONE Solution!Have ONLY ONE Solution!
Inequalities:Inequalities:Have MANY Solutions!!!!!Have MANY Solutions!!!!!
Write equations and inequalities
EXAMPLE 1
a. The difference of twice a number k and 8 is 12.
b. The product of 6 and a number n is at least 24.
6n ≥ 24
Verbal Sentence Equation or Inequality
2k – 8 = 12
GUIDED PRACTICE for Example 1
d. Write an equation or an inequality: The quotient of a number p and 12 is at least 30.
ANSWER
P12
>– 30
c. A number y is no less than 5 and no more than 13.
5 ≤ y ≤ 13
Check possible solutions
EXAMPLE 2
Check whether 3 is a solution of the equation or inequality.
a. 8 – 2x = 2
c. 2z + 5 > 12
b. 4x – 5 = 6
d. 5 + 3n ≤ 20
4(3) – 5 6?=
8 – 2(3) 2?=
2(3) + 5 12>?
5 + 3(3) 20
≤?
Equation/Inequality Substitute Conclusion
2 = 2 3 is a solution.
14 ≤ 20 3 is a solution.
7 = 6 3 is not a solution.
X
11 > 12 3 is not a solution.
X
GUIDED PRACTICE for Example 2
Check to see whether or not 5 is a solution of the equation or inequality.
a. 9 – x = 4
b. b + 5 < 15
9 – 5 4?=
Equation/Inequality Substitute Conclusion
4 = 4 5 is a solution.
5 + 5 15<?
10<15 5 is a solution.
c. 2n + 3 21>– 13 215 is NOT a solution.
>–2(5) + 3 21>–?
X
EXAMPLE 3Use mental math to solve an equation
Equation
a. x + 4 = 10
b. 20 – y = 8
c. 6n = 42
a5
=9d.
Think
What number plus 4 equals 10?
20 minus whatnumber equals 8?
6 times what numberequals 42?What number divided by 5 equals 9?
Solution
6
12
7
Check
20 –12 = 8
6 + 4 = 10
6(7) = 42
455 =9 45
GUIDED PRACTICE for Example 3
Solve the equation using mental math.
Equation
5. m + 6= 11
6. 5x = 40
Think
What number plus 6 equals 11?
5 times whatnumber equals 40?What number dividedby 4 equals 10
Solution
5
8
40
Check
5(8) = 40
5 + 6 = 11
40 = 10 4
7. r = 104
Is 2 a solution to 4z – 5 < 3?Is 2 a solution to 4z – 5 < 3?
Answer NowAnswer Now
1.1. A solutionA solution
2.2. NOT a solutionNOT a solution
Solve a multi-step problemEXAMPLE 4
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. What is the regular price of 4 tickets? If you pay the regular price this time and share it equally, how much does each person pay?
Mountain Biking
Solve a multi-step problem
EXAMPLE 4
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
Solve a multi-step problemEXAMPLE 4
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. •Answer: The regular price for 4 tickets is $27.
Step 3: Find Cost Per Person
$27 / 4 people = 6.75
•Answer: $6.75 per person.
GUIDED PRACTICE for Examples 4 and 5
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?
GUIDED PRACTICE for Examples 4 and 5
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
GUIDED PRACTICE for Examples 4 and 5
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
Write and check a solution of an inequality
EXAMPLE 5
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. 18(20) = 360 360 > 351
Answer: An average of 20 points per game will be enough.
GUIDED PRACTICE for Examples 4 and 5
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) = 352 352 > 351So, 22 is a solution.