Solving & Graphing Inequalities

Solving & Graphing Inequalities

Properties of Inequality

Opening

Properties of Inequality

You have been using the addition and multiplication properties of equality to help you solve equations.

Addition Property of Equality
You can add any number to both sides of an equation and the result will be a true equation.

Multiplication Property of Equality
You can multiply both sides of an equation by any number and the result will be a true equation.

Today you will work with inequalities. For the inequality –3 < 4, discuss what happens when you add or multiply as specified.

  • Add 2 to each side.
  • Add –2 to each side.
  • Multiply each side by 2.
  • Multiply each side by –2.

Math Mission

Opening

Discover how the processes for solving inequalities and solving equations are similar and different.

Numeric Inequalities

Work Time

Numeric Inequalities

Again start with –3 < 4. For each problem, predict if the inequality will remain true. Then verify your predictions.

  • Subtract 2 from both sides.
  • Subtract –2 from both sides.
  • Divide both sides by 2.
  • Divide both sides by –2.

Algebraic Inequalities

Work Time

Algebraic Inequalities

Look at this solution.

2x > 14

x > 7

  • Explain each step in finding the solution.
  • Graph the inequality on the number line.
  • Check your work by using values from the number line in the original inequality.

INTERACTIVE: Graph an Inequality

Hint:

Think about whether the graph should have a closed circle or an open circle.

Justify Steps and Graph

Negative Coefficients

Work Time

Negative Coefficients

Look at this solution.

−4x + 2 > 10

−4x > 8

x < −2

  • Explain each step in finding the solution.
  • Graph the inequality on the number line.
  • Check your work by using values from the number line in the original inequality.

HANDOUT: Negative Coefficients
INTERACTIVE: Graph an Inequality

Fractions in Inequalities

Work Time

Fractions in Inequalities

Look at this solution.

12x+4112x5x10

 

  • Explain each step in finding the solution.
  • Graph the inequality on the number line.
  • Check your work by using values from the number line in the original inequality.

HANDOUT: Fractions in Inequalities
INTERACTIVE: Graph an Inequality

 

Prepare a Presentation

Work Time

Prepare a Presentation

  • Summarize your learning about working with inequalities.
  • Support your conclusions with examples of your work.

Challenge Problem

Solve this inequality:

  • 3x75x+12

Hint:

Think about how addition and subtraction are related and how multiplication and division are related.

Make Connections

Performance Task

Ways of Thinking: Make Connections

Take notes about your classmates' explanations for solving each inequality and summarize them into the properties of inequalities.

Hint:

As your classmates present, ask questions such as:

  • What does solving an inequality mean?
  • Why did you change the direction of the sign?
  • How did you decide whether or not to fill in the circle on the number line?
  • If the inequality symbol was an equals sign, how would your answer be similar? How would it be different?

Solve Inequalities

Work Time

Apply the Learning: Solve Inequalities

Addition Property of Inequality
You can add any number to both sides of an inequality and the result will be true.

Multiplication Property of Inequality
You can multiply both sides of an inequality by any positive number and the result will be true. If you multiply both sides of an inequality by a negative number, you will need to reverse the direction of the inequality sign in order for the inequality to be true.

Solve each inequality and represent on a graph. Check your solution by checking values.

  1. x + 6 > 3
  2. −4x ≥ 12
  3. 4 − 2x < 12
  4. 3x + 4 ≥ 15

INTERACTIVE: Graphing Inequalities

Hint:

Think about how you can check your work using the solution on the number line.

Properties of Inequality

Formative Assessment

Summary of the Math: Properties of Inequality​

Write a summary of what you have learned about properties of inequality.

Hint:

Do you explain how solving inequalities is similar to and different from solving equations?

Reflect On Your Work

Work Time

Reflection

Write a reflection about the ideas discussed in class today. Use the sentence starter below if you find it to be helpful.

Something new I learned today is...