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Solving Inequalities by Addition and Subtraction

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Solving Inequalities by Addition and SubtractionRecall 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.


Addition and Subtraction Properties of Inequalities

When you add or subtract the same value from each side of an inequality, the inequality remains true.

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Answer: The solution is the set {all numbers greater than 77}.

Solve Then check your solution.Original inequalityAdd 12 to each side.This means all numbers greater than 77.

Check Substitute 77, a number less than 77, and anumber greater than 77.

Solve by Adding

Solve Then check your solution.

Answer: or {all numbers less than 14}

Solve by Adding


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.

Solve Then graph it on a number line.Original inequalityAdd 9 to each side.Simplify.

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

The dot at 21 shows that 21 is included in the inequality.

The heavy arrow pointing to the left shows that the inequality includes all the numbers less than 21.

Graph the Solution

Solve Then graph it on a number line.

Answer:

Graph the Solution

Solve Then graph the solution.Original inequalitySubtract 23 from each side.Simplify.

Answer: The solution set is

Solve by Subtracting

Solve Then graph the solution.

Answer:

Solve by Subtracting


Terms with variables can also be subtracted from each side to solve inequalities.

Then graph the solution.Original inequalitySubtract 12n from each side.Simplify.

Answer: Since is the same as the solution set is

Variables on Both Sides

Then graph the solution.

Answer:

Variables on Both Sides


The table below shows some common verbal phrases and the corresponding mathematical inequalities.

Inequalities< >

• is less than

• is fewer than

• is greater than

• is more than

• exceeds

• is less than or equal to• is no more than• is at most

• is greater than or equal to• is no less than

• is at least

Verbal problems containing phrases like greater than or less than can often be solved by using inequalities.

Write an inequality for the sentence below. Then solve the inequality.Seven times a number is greater than 6 times that number minus two.

Seven timesa number

is greaterthan

six timesthat number minus two.

7n 6n 2> –

Simplify.Subtract 6n from each side.Original inequality

Answer: The solution set is

Write and Solve an Inequality

Let n = the number

Write an inequality for the sentence below. Then solve the inequality.Three times a number is less than two times that number plus 5.

Answer:

Write and Solve an Inequality

Let n = the number

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.

Inequality 100

The total costis less thanor equal to $100.

Write an Inequality to Solve a Problem

Solve the inequality.

Answer: The second pass must cost no more than $45.01.

Original inequalitySubtract 54.99 from each side.Simplify.


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


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