1.4 Equations and Inequalities
What you should learn
GOAL
GOAL
1
Check solutions and solve equations using
mental math.
2
Check solutions of inequalities in a real-life
problems, such as regulating your cat’s
caloric intake.
Why you should learn it
To solve a real-life problem such as how long you
must save money to buy a violin.
1.4 Equations and Inequalities
GOAL
1
CHECKING AND SOLVING EQUATIONS
VOCABULARY
•equation— formed when an equal sign is placed
between two expressions
•open sentence— an equation that contains at least one
variable
•solution of an equation— a number which will make a
true statement when substituted for the
variable in a single-variable equation
Checking a possible solution
To check if a number is a possible solution to a variable
equation, simply substitute the number for the variable. If
the statement is true, the number is a solution. If the
statement is false, the number is not a solution.
EXAMPLE 1
EXAMPLE 2
Extra Example 1
Check whether the numbers 3 and 4 are solutions of the
equation 5x – 7 = 8.
x = 3: 5x – 7 = 8
x = 4: 5x – 7 = 8
5(3) – 7 = 8
5(4) – 7 = 8
8=8
13 ≠ 8
3 is a solution.
4 is not a solution.
Extra Example 2
Check whether the numbers 5, 10, and 15 are solutions of
the equation 12s + 5 = 125.
x = 5:
x = 10:
x = 15:
12s + 5 = 125
12s + 5 = 125
12s + 5 = 125
12(5) + 5 = 125
12(10) + 5 = 125
12(15) + 5 = 125
65 ≠ 125
125 = 125
185 ≠ 125
Only 10 is a solution to the equation.
VOCABULARY
• solving an equation— finding all the solutions of an
equation
For now, we will solve equations using mental math.
ACTIVITY
Developing Using Mental Math to Solve Equations
Concepts
Please work through the Activity on page 25.
EXAMPLE 3
Extra Example 3
You need to buy ingredients to make a pizza. Cheese costs
$5.99, pizza dough costs $2.39, tomato sauce costs
$2.49, and pepperoni costs $2.98. You have a twentydollar bill. About how much change will you receive?
Work this out in the best calculator you have—your brain!
Hint: Round each amount to the nearest half-dollar.
Possible solution:
Rounding each amount gives $6 + $2.50 + $2.50 + $3 = $14.
Subtracting gives $20 – $14 = $6.
You will receive about $6 in change.
Checkpoint
1. Check whether the numbers 3, 6, and 9 are solutions of the
equations 8s – 20 = 52.
9 is a solution; 3 and 6 are not.
2. Use mental math to solve: You need to buy supplies for
school. A box of pencils costs $2.98, a package of pens
costs $3.95, a notebook costs $4.49, and a ruler costs
$0.89. You have $10. How much more money do you
need?
You need about $2.50.
1.4 Equations and Inequalities
GOAL
2
CHECKING SOLUTIONS OF INEQUALITIES
VOCABULARY
•inequality— formed when an inequality symbol is
placed between two expressions
•solution of an inequality— a number which will make a
true statement when substituted for the
variable in an inequality
Inequality Symbols
Do you know how to read each of the four inequality symbols?
Let’s find out! Match each symbol with its meaning.
<
is greater than or equal to
>
is less than
≤
is less than or equal to
≥
is greater than
Click to see the answers.
To check if a number is a solution of an inequality,
simply substitute the number for the variable. If it
makes a true statement, the number is a solution.
EXAMPLE 4
EXAMPLE 5
Extra Example 4
Decide whether 3 is a solution of the inequality.
a. 3x – 2 > 5
3(3)  2  5
75
YES
c. x – 1 ≥ 2
3 1 2
22
b. x + 5 < 8
35 8
88
NO
d. 4x2 ≤ 32
4  32  32
YES
36  32
NO
Extra Example 5
If you weigh 120 lbs, the number of calories you will burn
while walking briskly is about 240 times the number of hours
you walk. Using the following values of x, will a 120 lb
person walking for x hours burn at least 500 calories?
a. 2 hours
b. 3 hours
Hint: First, write a verbal model, then the inequality.
Model:
Calories burned ≥ 500
Labels/Inequality: 240 • number of hours ≥ 500
240 • t ≥ 500
Solution:
a. 240t ≥ 500
240(2) ≥ 500
480 ≥ 500 NO
b. 240t ≥ 500
240(3) ≥ 500
720 ≥ 500 YES
Checkpoint
1. Decide whether 4 is a solution of the inequality.
a. 2x – 1 > 5
c. x + 3 ≥ 5
YES
YES
b. x – 2 > 6 NO
d. 3x2 ≤ 52 YES
2. Your scores on the last three tests were 82, 87, and 74.
Your score on the next test is x points. Do the following
values for x give you at least 320 total points?
a. 75 NO
b. 81 YES
QUESTIONS?