In this lesson, students see more examples of inequalities in a context. This time, many inequalities involve negative coefficients. This illustrates the idea 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, or reasoning about number lines. All of these techniques exemplify MP1: making the problem more concrete and visual and asking “Does this make sense?” In this lesson, the emphasis is on reasoning about the context.
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 either think about the context or 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.
- Interpret inequalities that represent situations with a constraint.
- Solve an equation of the form $px+q=r$ to determine the boundary point for an inequality of the form $px+q \gt r$ or $px+q \lt r$.
- Use substitution or reasoning about the context to justify (orally and in writing) whether the values that make an inequality true are greater than or less than the boundary point.
Let’s solve more complicated inequalities.
Several activities suggest providing students with blank number lines to use for scratch work. One way to accomplish this is to print a line with unlabeled, evenly-spaced tick marks, and place these into sheet protectors. Students can write on these with dry erase markers and wipe them off.
- I can describe the solutions to a inequality by solving a related equation and then reasoning about values that make the inequality true.
- I can write an inequality to represent a situation.
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.