1. 1.4 Write Equations And Inequalities
Algebra I
1.4 Write Equations And Inequalities

2. VOCAB
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

3. VOCAB
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

4. 1.4 Write Equations and Inequalities
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

5. 1.4 Write Equations and Inequalities
The BIG Difference
Equations:
Have ONLY ONE Solution!
Inequalities:
Have MANY Solutions!!!!!

6. EXAMPLE 1
Write equations and 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

7. c. A number y is no less than 5 and no more than 13.
GUIDED PRACTICE
for Example 1
c. A number y is no less than 5 and no more than 13.
5 ≤ y ≤ 13
d. Write an equation or an inequality: The quotient of a number p and 12 is at least 30.
ANSWER P 12
> – 30

8. Check possible solutions
EXAMPLE 2
Check possible solutions
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 3 is a solution. 
b. 4x – 5 = 6
4(3) – ? =
7 = 6 3 is not a solution. X
c. 2z + 5 > 12
2(3)
> ?
11 > is not a solution. X
d n ≤ 20
5 + 3(3) 20 ≤ ?
14 ≤ is a solution. 

9. GUIDED PRACTICE for Example 2
Check to see whether or not 5 is a solution of the equation or inequality.
Equation/Inequality
Substitute
Conclusion
a. 9 – x = 4
9 – ? =
4 = 4 5 is a solution. 
b. b + 5 < 15
< ?
10<15 5 is a solution. 
c. 2n
> –
2(5)
> – ?
is NOT a solution.
> – X

10. EXAMPLE 3
Use mental math to solve an equation
Equation
Think
Solution
Check
a. x + 4 = 10
What number plus 4 equals 10? 6
6 + 4 = 10 
b – y = 8
20 minus what number equals 8? 12
20 –12 = 8 
c. 6n = 42
6 times what number equals 42? 7
6(7) = 42  a 5 = 9 d. 45 5 =
9 
What number divided by 5 equals 9? 45

11. GUIDED PRACTICE
for Example 3
Solve the equation using mental math.
Equation
Think
Solution
Check
5. m + 6= 11
What number plus 6 equals 11? 5
5 + 6 = 11 
6. 5x = 40
5 times what number equals 40? 8
5(8) = 40 
7. r = 10 4
What number divided by 4 equals 10 40
40 = 10  4

12. Is 2 a solution to 4z – 5 < 3?
A solution
NOT a solution
Answer Now

13. Solve for f: 12 24 4
Answer Now

14. EXAMPLE 4
Solve a multi-step problem
Mountain Biking
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?

15. Step 1: Write a verbal model.
EXAMPLE 4
Solve a multi-step problem
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

16. Step 2: Use mental math to solve the equation p – 10 = 17.
EXAMPLE 4
Solve a multi-step problem
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.

17. 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?

18. STEP 1: Write a verbal model.
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

19. STEP 2: Use mental math to solve the equation p – 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

20. Write and check a solution of an inequality
EXAMPLE 5
Write and check a solution of an inequality
You play 18 games of basketball. Last year you scored over 351. What was the average number of points you earned for each game.
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.

21. 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 > 351
So, 22 is a solution.