Finding Solutions of Inequalities

Getting Started

The student misinterprets the meaning of the inequality.

The student:

• Treats the inequality as if it is an equation.
• Finds one value that makes the statement true but does not check other values.
• Misinterprets the inequality symbol, particularly when the variable is to its right.

What does this symbol mean? What does it tell you about the relationship between the quantities it is comparing?

I see you found one solution for this inequality. Is it possible that there are more?

The variable is on the right-hand side of this inequality symbol. Can you rewrite the inequality in an equivalent form but with the variable on the left?

Review the meaning of the inequality symbols , and what it means for a value to be a solution of an inequality. Given an inequality, provide examples of values that are solutions as well as values that are not solutions. Demonstrate how to use substitution to test values to determine whether or not they are solutions. Provide additional inequalities and sets of values and ask the student to determine if any are solutions. Assist the student in using the order of operations to evaluate expressions within the inequality and to correctly interpret the meaning of the inequality symbol. Guide the student to carefully and systematically check each value and to write the appropriate corresponding work on the paper.

If needed, assist the student in correctly interpreting the meaning of inequalities in which the variable is on the right side of the symbol, such as 5 > x. Guide the student to correctly read the inequality as “five is greater than x” and to consider what that means about x in relationship to the number five. Then show the student how to rewrite the inequality in an equivalent form (e.g., as x < 5). Encourage the student to rewrite inequalities in this form if he or she is having difficulty understanding the inequality as written.

Moving Forward

The student makes errors when calculating with rational numbers.

The student makes errors with:

• Integer operations (e.g., writing 16 0 = 16).
  
  
  
  
• Decimal operations (e.g., writing 3.1 + 9 = 4.0).
  
  
  
  
• Fraction operations (e.g., writing 16 10= 16   10 = 8 10 = 80). 

What happens when you multiply a number by zero?

Can you explain how you added these decimals?

What does it mean to multiply a mixed number and a whole number? Is that the same as multiplying just the fraction portion times the whole number?

Provide direct feedback on any errors that the student might have made and allow the student to correct them. Review operations with rational numbers as well and address the specific issue the student is having.

Provide additional opportunities to work with rational numbers in the context of solving equations and inequalities.

Almost There

The student makes minor errors in calculation or in interpreting results of calculations.

The student:

• Makes a minor mathematical error in a step of work.
• Reverses the meaning of a strict and non-strict inequality symbol. 
  
  
  
  
• Has inadequate overall work to support answers (e.g., only shows work for the values that are solutions but no work or explanation for values that are not).
  
     

I think you made a small error. Can you go back to check your work?

What symbol do we use for equality? Which inequality symbol incorporates part of that? How are they related?

How did you decide that these values are not solutions? Can you show work to justify that decision?

Have the student review his or her work and correct any errors that were made. If needed, assist the student in locating the errors and provide feedback that directly addresses them.

Model showing work appropriately and ask the student to revise his or her work accordingly.

Got It

The student provides complete and correct responses to all components of the task.

The student shows work to justify that:

• None of the given values are solutions of the first inequality. The student may also explain that no value will ever make the statement true because adding four to a number will always be more than subtracting 12 from the same number. 
• 3.1 and 4.2 are solutions of the second inequality, while 5.3 and 6.4 are not.
• 0, and are solutions of the third inequality, while 10 is not. The student may show work in fraction or decimal form.

You said that none of the given values of the first inequality are solutions, but are there values that were not given that could be solutions? Why or why not?

Are there values that were not given that could be solutions of the second inequality? Why or why not?

Ask the student to solve each inequality and then graph its solutions. Guide the student to consider how many solutions linear inequalities might have.

Challenge the student to consider the meaning of compound inequalities such as x + 5 < 12 and 3x > -9. Ask the student to find several examples of values that are solutions and several examples of values that are not solutions.