Algebra I
1.4 Write Equations
And Inequalities
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
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
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
1.4 Write Equations and
Inequalities

The BIG Difference


Equations:
 Have ONLY ONE Solution!
Inequalities:
 Have MANY Solutions!!!!!
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
GUIDED PRACTICE
for Example 1
c. A number y is no less than 5 and
no more than 13.
d.
5 ≤ y ≤ 13
Write an equation or an inequality: The quotient of a
number p and 12 is at least 30.
ANSWER
P > 30
–
12
EXAMPLE 2
Check possible solutions
Check whether 3 is a solution of the equation or
inequality.
Equation/Inequality Substitute
Conclusion
a. 8 – 2x = 2
? 2
8 – 2(3) =
b. 4x – 5 = 6
2=2
3 is a solution.
4(3) – 5 =? 6
c. 2z + 5 > 12
7 = 6X
3 is not a solution.
2(3) + 5 ? 12 11 > 12X
>
3 is not a solution.
d. 5 + 3n ≤ 20
?
5 + 3(3) ≤ 20


14 ≤ 20
3 is a solution.
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 – 5 ?=
b. b + 5 < 15
?
5 + 5 < 15
c. 2n + 3 >
– 21
?
X
2(5) + 3 >
21
– 21 13 >
–
5 is NOT a solution.
4

4=4
5 is a solution.

10<15
5 is a solution.
EXAMPLE 3
Equation
a.
Use mental math to solve an equation
Think
x + 4 = 10 What number
Solution
Check
6
6 + 4 = 10 
20 minus what
number equals 8?
12
20 –12 = 8 
6 times what
number
equals 42?
What number
divided by 5 equals
9?
7
6(7) = 42 
45
45
=9 
5
plus 4 equals 10?
b. 20 – y = 8
c.
6n = 42
d.
a =9
5
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
Is 2 a solution to 4z – 5 < 3?
1.
2.
A solution
NOT a solution
Answer Now
Solve for f:
f
4
6
1.
2.
3.
4.
0
12
24
4
Answer Now
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?
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
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.
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
EXAMPLE 5
Write and check a solution of an inequality
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 > 351
So, 22 is a solution.