# Lesson 19

Solutions to Inequalities in One Variable

### Lesson Narrative

In this lesson, students revisit the meaning of the solutions to an inequality in one variable and recall that the solution set is a range of values. They also investigate different ways to find the solution set to an inequality—by reasoning about the quantities and relationships in context, by guessing some values, substituting them into the inequality, and checking them to see if they make an inequality true, and by first solving a related equation in one variable. Along the way, students reason abstractly and quantitatively (MP2).

Two optional activities are included in this lesson. The first is to give students an additional opportunity to make sense of the solutions to an inequality in terms of a situation. The second optional activity introduces them to graphing two-variable equations as a way to find solutions to one-variable inequalities.

Later, students will use the understanding they build here to solve more sophisticated problems and to find solutions to linear inequalities in two variables.

### Learning Goals

Teacher Facing

• Find the solution to a one-variable inequality by reasoning and by solving a related equation and testing values greater than and less than that solution.
• Graph the solution to an inequality as a ray on a number line and interpret the solution in context.
• Understand that the solution to an inequality is a range of values that make the inequality true.

### Student Facing

• Let’s find and interpret solutions to inequalities in one variable.

### Required Preparation

Devices that can run Desmos (recommended) or other graphing technology are needed for the optional activity, More or Less? The digital version with an embedded applet is recommended for all classes.

### Student Facing

• I can graph the solution to an inequality in one variable.
• I can solve one-variable inequalities and interpret the solutions in terms of the situation.
• I understand that the solution to an inequality is a range of values (such as $x>7$) that make the inequality true.

Building On