Transcript Slide 1
Solving Inequalities by Addition and Subtraction
m 8425 69 23
y 12 x
y 3x 5
y5
Solving Inequalities by Addition and Subtraction
Recall that statements with greater than (>), less than
(<), greater than or equal to (≥) or less than or equal to
(≤) are inequalities.
Solving an inequality means finding values for the
variable that make the inequality true.
You can solve inequalities by using the Addition and
Subtraction Properties of Inequalities.
Solving Inequalities by Addition and Subtraction
Addition and Subtraction Properties of Inequalities
When you add or subtract the same value from
each side of an inequality, the inequality
remains true.
24
63
23 43
57
6 4 3 4
2 1
Solve by Adding
Solve
Then check your solution.
Original inequality
Add 12 to each side.
This means all numbers
greater than 77.
Check Substitute 77, a number less than 77, and a
number greater than 77.
Answer: The solution is the set
{all numbers greater than 77}.
Solve by Adding
Solve
Then check your solution.
Answer:
or {all numbers less than 14}
Solving Inequalities by Addition and Subtraction
The solution of the inequality in Example 1 was
expressed as a set.
A more concise way of writing a solution set is to use
set-builder notation.
The solution in set-builder notation is {k | k < 14}.
This is read as the set of all numbers k such that k is
less than 14.
The solutions can be represented on a number line.
Graph the Solution
Solve
Then graph it on a number line.
Original inequality
Add 9 to each side.
Simplify.
Answer: Since
is the same as y 21,
the solution set is
The heavy arrow pointing
to the left shows that the
inequality includes all the
numbers less than 21.
The dot at 21
shows that 21
is included in
the inequality.
Graph the Solution
Solve
Answer:
Then graph it on a number line.
Solve by Subtracting
Solve
Then graph the solution.
Original inequality
Subtract 23 from each side.
Simplify.
Answer: The solution set is
Solve by Subtracting
Solve
Answer:
Then graph the solution.
Solving Inequalities by Addition and Subtraction
Terms with variables can also be
subtracted from each side to solve
inequalities.
Variables on Both Sides
Then graph the solution.
Original inequality
Subtract 12n from each side.
Simplify.
Answer: Since
is the same as
solution set is
the
Variables on Both Sides
Then graph the solution.
Answer:
Solving Inequalities by Addition and Subtraction
Verbal problems containing phrases like greater than
or less than can often be solved by using inequalities.
The table below shows some common verbal phrases and the
corresponding mathematical inequalities.
Inequalities
<
• is less than
• is fewer than
>
• is greater than • is less than or • is greater than
equal to
or equal to
• is more than
• is no more
• is no less than
than
• exceeds
• is at least
• is at most
Write and Solve an Inequality
Write an inequality for the sentence below. Then
solve the inequality.
Seven times a number is greater than 6 times that number
minus two.
Let n = the number
Seven times is greater
six times
a number
than
that number minus
two.
7n
>
6n
–
2
Original inequality
Subtract 6n from each side.
Simplify.
Answer: The solution set is
Write and Solve an Inequality
Write an inequality for the sentence below. Then
solve the inequality.
Three times a number is less than two times that number
plus 5.
Let n = the number
Answer:
Write an Inequality to Solve a Problem
Entertainment Alicia wants to buy season passes to
two theme parks. If one season pass cost $54.99,
and Alicia has $100 to spend on passes, the second
season pass must cost no more than what amount?
Words
The total cost of the two passes must be
less than or equal to $100.
Variable
Let
Inequality
the cost of the second pass.
is less than
The total cost
or equal to
$100.
100
Write an Inequality to Solve a Problem
Solve the inequality.
Original inequality
Subtract 54.99 from
each side.
Simplify.
Answer: The second pass must cost no more than $45.01.
Write an Inequality to Solve a Problem
Michael scored 30 points in the four rounds of the
free throw contest. Randy scored 11 points in the
first round, 6 points in the second round, and 8 in
the third round. How many points must he score in
the final round to surpass Michael’s score?
Answer: 6 points