In this activity, students move from writing individual values that satisfy conditions for a real-world situation to writing inequalities that describe the conditions. The context of nuts and bolts is used to emphasize the importance of precision and give students a sense of how error tolerance is used in the real world. 

The purpose of the discussion is to make a strong connection between the context, a range of distances on a number line, and inequalities involving absolute value.

Invite groups to share their inequalities that have absolute values. Then ask students,

• “What is the maximum possible thread length that is acceptable?” (1.1 millimeters)
• “What is the minimum possible thread length that is acceptable?” (0.9 millimeter)
• “A different type of bolt should have a thread length of 0.5 millimeter, but it is still acceptable if the threads are within 0.03 millimeter of that value. What is an inequality that fits this situation?” ( )
• “What would a number line that shows the solutions for the inequality you wrote in the last problem look like?” (An open circle at 0.9 millimeter with shading to the left and an open circle at 1.1 millimeters with shading to the right.)

The purpose of the discussion is for students to understand how to solve inequalities that include absolute values. 

Invite groups to share their solutions for the four inequalities. Ask students,

• “How do the first two inequalities compare to the last two?” (The first two involve the absolute values being less than a value, and the last two are greater than a value.)
• “How do the solutions on the number line compare for the first two inequalities and the last two inequalities?” (The first two number lines are shaded between 2 values. The last two number lines are shaded on the outside of 2 values.)
• "Does having the absolute value on the right side of the inequality change how you think about it?" (It changes how I think about it a little bit. I think about it as 3 being greater than the distance between and -3 rather than the distance between and -3 being less than 3.)
• “If these inequalities represent the distance you should be from an object, how would you describe the solution for the first and last problems?” (In the first one, we need to be closer than 1 unit away from an object that is at a position of 5 units. In the last one, we need to be farther away than  unit from the object that is at a position of 10 units.)
• “How does the solution to the first inequality change if it is written as ?” (It does not change the solution.)