Day 2: Solving One-Variable Inequalities
Warm-Up
Graph the inequality y < -3 and express in interval notation.
Solving Inequalities Using Addition and Subtraction
Solving inequalities with addition and subtraction is just like solving equations.
Exercise
1) −6 + 𝑛 > −3
Find the solution set of each inequality. Graph and express
in interval notation and set builder notation
Set-builder Notation:
Interval Notation:
2) −5 ≤ −3 + 𝑦
Set-builder Notation:
Interval Notation:
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Solving Inequalities with Multiplication and Division
As you saw, solving inequalities using addition and subtraction is just like solving
equations. However, when it comes to multiplication and division, there is one slight
difference.
This difference involves multiplication and division by negative numbers.
Consider this example. What inequality sign belongs in the box?
3<5
___ ∙ 3 < 5 ∙ _______
_____
_______
When multiplying or dividing both sides of an inequality by a negative number,
the sense of the inequality changes.
The “sense” of the inequality refers to the statement that the inequality is making.
Note: We do not change the sense of the inequality when we reverse the entire
statement. We simply change its direction.
For example:
4 > 3 and 3 < 4
These inequalities are the same.
We did not change the sense of the inequality.
4 > x and x < 4
These inequalities are the same.
We did not change the sense of the inequality.
x < 4 and x > 4
These inequalities are different.
One says that x is greater than 4 and one
says it is less.
We DID change the sense of the inequality.
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Model Problems
Solve and graph each inequality. Write the solution in setbuilder and interval notation.
1) 2𝑥 − 6 ≥ 4
Set-builder Notation:
Interval Notation:
2) 2(4 − 𝑥) > 4
Set-builder Notation:
Interval Notation:
2
3) − 𝑥 + 5 < −3
3
Set-builder Notation:
Interval Notation:
4)
4≤
𝑘−5
−2
Set-builder Notation:
Interval Notation:
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Exercise
Solve and graph each inequality. Write the solution in set-builder
and interval notation.
2
1) 𝑥 < 4
Set-builder Notation:
3
Interval Notation:
3
2) − (𝑥 + 1) ≥ 3
4
Set-builder Notation:
Interval Notation:
3)
5 > 12𝑎 − (3 − 4𝑎)
Set-builder Notation:
Interval Notation:
4)
1 ≤ −3 +
𝑦+6
2
Set-builder Notation:
Interval Notation:
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Challenge Express each phrase using an inequality.
x is a positive number _____________ x is not a negative number _____________
x is a negative number _______________
Homework
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