# ch 6 1 solving inequalities add and subtract

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```					Solving Inequalities by Addition and Subtraction

m   8 2
 4 5      6
9   2
3

y1x
2

y  3x  5
y5
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.
24                     63
23 43                  6 4  3 4
57                     2  1
Original inequality
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}.

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
Simplify.
Answer: Since         is the same as y  21,
the solution set is

The heavy arrow pointing               The dot at 21
to the left shows that the             shows that 21
inequality includes all the            is included in
numbers less than 21.                  the inequality.
Graph the Solution
Solve      Then graph it on a number line.

Solve by Subtracting
Solve              Then graph the solution.
Original inequality
Subtract 23 from each side.
Simplify.
Solve by Subtracting
Solve          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              the
solution set is
Variables on Both Sides
Then graph the solution.

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 greater than • is less than or   • is greater than
equal to            or equal to
• is fewer than   • is more than    • is no more        • is no less than
than
• exceeds         • is at most        • is at least
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.
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
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     the cost of the second pass.
is less than
The total cost    or equal to      \$100.
Inequality                                              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?