# Lesson 15

Efficiently Solving Inequalities

### Lesson Narrative

In this lesson, students see more examples of inequalities. This time, many inequalities involve negative coefficients. This reinforces the point that solving an inequality is not as simple as solving the corresponding equation. After students find the boundary point, they must do some extra work to figure out the direction of inequality. This might involve reasoning about the context, substituting in values on either side of the boundary point, and reasoning about number lines. All of these techniques exemplify MP1: making the problem more concrete and visual and asking, “Does this make sense?”

It is important to understand that the goal is not to have students learn and practice an algorithm for solving inequalities like “whenever you multiply or divide by a negative, flip the inequality.” Rather, we want students to understand that solving a related equation tells you the lower or upper bound of an inequality. To know whether values greater than or less than the boundary number make the inequality true, it's best to test some values that are above and below the boundary number. This way of reasoning about inequalities will serve students well long into their future studies, whereas students are very likely to forget a procedure memorized for a special case.

### Learning Goals

Teacher Facing

• Compare and contrast (orally) solutions to equations and solutions to inequalities.
• Draw and label a graph on the number line that represents all the solutions to an inequality.
• Generalize (orally) that you can solve an inequality of the form $px+q \gt r$ or $px+q \lt r$ by solving the equation $px+q=r$ and then testing a value to determine the direction of the inequality in the solution.

### Student Facing

Let’s solve more complicated inequalities.

### Student Facing

• I can graph the solutions to an inequality on a number line.
• I can solve inequalities by solving a related equation and then checking which values are solutions to the original inequality.

Building Towards

### Glossary Entries

• solution to an inequality

A solution to an inequality is a number that can be used in place of the variable to make the inequality true.

For example, 5 is a solution to the inequality $$c<10$$, because it is true that $$5<10$$. Some other solutions to this inequality are 9.9, 0, and -4.