This Warm-up activates what students know about the solutions to an inequality and ways to find the solutions. For the first time, students refer to the solutions as the solution set of the inequality. Throughout their work with one-variable inequalities, students will use the terms “solutions” and “solution set” interchangeably.

By now, students are likely to have internalized that a solution to an equation in one variable is a value that makes the equation true. The work here makes it explicit that we can extend this understanding of “solution” to inequalities in one variable.

Some students may have been taught to solve inequalities the same way we solve equations, with the added rule along the lines of "flip the symbol when dividing or multiplying both sides of an inequality by a negative number." They may or may not have understood why the rule is the way it is. (If a student shares this method, emphasize that they will look at different strategies in the lesson and adopt whichever ways that they can explain or justify.)

If students approach the last question (finding a solution to ) by performing operations directly on the inequality but neglect to reverse the inequality symbol, they would find solutions that result in false statements. Use this opportunity to point out that we may run into problems with this method. It is not essential to discuss why or to suggest better approaches at this point. There will be other opportunities in this lesson to reason about the solutions and to witness the same issue (in the second non-optional activity—“Equality and Inequality”).

Invite students to share some solutions to the first inequality and explain what it means for a value to be a solution to . Be sure to mention some negative numbers that are solutions. If necessary, show the solution set on a number line. Then, focus the discussion on the second inequality.

Ask each student to mark the one solution they have for on the class number line (from the Launch). (Students could draw a point or put a dot sticker, if available, on the number line.) Discuss with students:

• "How did you know that the value you chose is a solution?" (When substituted for in the inequality, the value makes a true statement.)
• "What do you notice about all the points that are on the line?" (They are all to the left side of 1.)
• "On the number line, we can see that the solutions are values that are less than 1. All these values form the solution set to the inequality. Is there a way to write the solution set concisely, without using the number line and without writing out all the numbers less than 1?" (We can write ).
• "Does the solution set have anything to do with the solution to the equation ?" (The solution to the equation .)
• "Why does the solution set to the inequality involve numbers less than 1?" (The inequality can be taken to say "7 times is greater than 14." For the inequality to be true, must be greater than 2. For to be greater than 2, must be less than 1.)

Highlight that we can use a number line to concisely show the solution set to an inequality, but we can also write another inequality that shows the same information.

This activity encourages students to interpret an inequality, a related equation, and its solution set in terms of a situation. A context can help students intuit why the solutions to an inequality form a ray on the number line.

Students may recall this way of solving inequalities from middle school, but they may also solve by testing different possible values, or by reasoning about the relationship between quantities in other ways. The work here encourages students to reason quantitatively and abstractly (MP2), and to make sense of problems and persevere in solving them (MP1).

Monitor for students who use these strategies to find the solutions to :

• Try different values of until the inequality is no longer true.
• Solve the equation to find the number of students at which the costs for both options would be equal. That number is 17. Then, try a higher or lower number to see which side of the equation has a smaller value.

Plan to have students present in this order to support moving students toward more precise mathematical reasoning.

This optional activity gives students another opportunity to make sense, in context, of inequalities and their solutions. The context helps to reinforce why it makes sense for the solutions to an inequality to be represented by a set of points on one side of a particular value on the number line. It also reiterates the idea that solving a related equation can help us find that boundary value. In the context of part-time jobs presented here, that boundary value is a break-even point at which the two options are equal. On each side of that point, one job is more lucrative than the other.

Students are given an expression that represents the monthly pay for hours of work at each job. The expression for the pay at the restaurant is . Some students may notice that this expression could be rewritten as  and use this simpler equivalent expression throughout the activity. During the class discussion, invite students who use the simpler expression to share their rationale.

The purpose of this activity is to further develop the idea that we can solve an inequality by first solving a related equation. Previously, students reasoned about the solutions to an inequality in context. Here, they transition to solving an inequality without a context.

This activity offers another opportunity to point out the trouble with isolating the variable directly on the inequality statement, or with assuming that the solution set can be expressed using the same symbol as in the original inequality. (For example, if in the final step of solving, students go from to , they would end up with the wrong solution set.)

This optional activity allows students to visualize an inequality in one variable in another way—by graphing the expressions on each side and comparing the values of the expressions at different values of the variable. Doing so allows them to see the values at which the inequality is satisfied.

In the digital version of the activity, students use an applet to graph lines. The applet allows students to accurately graph lines and use a slider to compare values of the lines for the same -coordinate. Use the digital version if possible to allow students to focus on understanding the comparison.

If students don't have individual access, displaying the applet for all to see would be helpful during the launch.