In this Grade 9 Saxon Algebra 1 lesson, students learn to solve inequalities with variables on both sides by isolating the variable through addition, subtraction, multiplication, and division, including cases that require the distributive property and combining like terms. Students also identify special cases such as identities (always true) and contradictions (never true), and interpret solution sets using number line graphs. The lesson is part of Chapter 9 on Quadratic Functions and Equations and includes a real-world application connecting inequality solving to data trends.
Section 1
📘 Solving Inequalities with Variables on Both Sides
An inequality or equation that is always true is called an identity. A contradiction is an inequality or an equation that is never true (false).
Next, you’ll use familiar properties to solve these inequalities, including special cases that result in all or no solutions.
Section 2
Solving Inequalities with Variables on Both Sides
To solve an inequality with variables on both sides, the goal is to transform it by isolating the variable. Use the properties of inequality to move all variable terms to one side and all constant terms to the other side.
Think of it as a cosmic balance! To find the variable's true value, you must keep the inequality scale even. Whatever you do to one side—like adding or subtracting 7—you must do to the other. Systematically clear away terms until the variable is left standing all by itself, revealing its secret identity.
To solve , first add to both sides to get .
Then, subtract 6 from both sides to get .
Finally, divide by 6 to find the solution: .
Section 3
Example Card: Solving an Inequality with Variables on Both Sides
Getting variables on one side is the first step to taming these tricky inequalities.
Example Problem
Solve the inequality .
Section 4
Simplifying Each Side Before Solving
Before isolating the variable, you must first simplify each side of the inequality. Use the Distributive Property to remove parentheses and then combine any like terms to make the inequality as tidy as possible before you begin solving.
Don't jump into solving a messy problem! It's like trying to cook without prepping your ingredients. First, 'chop' expressions by distributing, then 'mix' the like terms together. Once each side is neat and organized, solving for the variable becomes much simpler and you are less likely to make a mistake.
For , first distribute to get .
Combine like terms on both sides to get .
Now it's a simple inequality ready to be solved.
Section 5
Example Card: Simplifying Each Side Before Solving an Inequality
Let's see how simplifying each side first makes solving this inequality much clearer.
Example Problem
Solve the inequality .
Book overview
Jump across lessons in the current chapter without opening the full course modal.
Continue this chapter
Expand to review the lesson summary and core properties.
Section 1
📘 Solving Inequalities with Variables on Both Sides
An inequality or equation that is always true is called an identity. A contradiction is an inequality or an equation that is never true (false).
Next, you’ll use familiar properties to solve these inequalities, including special cases that result in all or no solutions.
Section 2
Solving Inequalities with Variables on Both Sides
To solve an inequality with variables on both sides, the goal is to transform it by isolating the variable. Use the properties of inequality to move all variable terms to one side and all constant terms to the other side.
Think of it as a cosmic balance! To find the variable's true value, you must keep the inequality scale even. Whatever you do to one side—like adding or subtracting 7—you must do to the other. Systematically clear away terms until the variable is left standing all by itself, revealing its secret identity.
To solve , first add to both sides to get .
Then, subtract 6 from both sides to get .
Finally, divide by 6 to find the solution: .
Section 3
Example Card: Solving an Inequality with Variables on Both Sides
Getting variables on one side is the first step to taming these tricky inequalities.
Example Problem
Solve the inequality .
Section 4
Simplifying Each Side Before Solving
Before isolating the variable, you must first simplify each side of the inequality. Use the Distributive Property to remove parentheses and then combine any like terms to make the inequality as tidy as possible before you begin solving.
Don't jump into solving a messy problem! It's like trying to cook without prepping your ingredients. First, 'chop' expressions by distributing, then 'mix' the like terms together. Once each side is neat and organized, solving for the variable becomes much simpler and you are less likely to make a mistake.
For , first distribute to get .
Combine like terms on both sides to get .
Now it's a simple inequality ready to be solved.
Section 5
Example Card: Simplifying Each Side Before Solving an Inequality
Let's see how simplifying each side first makes solving this inequality much clearer.
Example Problem
Solve the inequality .
Book overview
Jump across lessons in the current chapter without opening the full course modal.
Continue this chapter