Getting Started 
Misconception/Error The student does not understand the meaning of absolute value. 
Examples of Student Work at this Level The student:
 Ignores the absolute value symbols and attempts to solve the inequalities as if the symbols were not there.
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 Performs operations on expressions within absolute value symbols.
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 Writes work that does not make mathematical sense.
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Questions Eliciting Thinking What does absolute value mean? How is it defined?
What is the absolute value of three? Of negative five? Of zero?
If the absolute value of some number is less than 4.7, what are some possible values of that number? 
Instructional Implications Review the concept of absolute value as distance from zero on a number line. Ask the student to identify all numbers with a given absolute value, for example,  ?  = 12. Have the student graph the two numbers (e.g., 12 and 12) on a number line and compare the results. Emphasize that since both 12 and 12 are 12 units from zero, they have the same absolute value. Model writing the absolute value of a number as two distinct values (rather than using the plus or minus notation). For example, write, â€śIf x = 12, then x = 12 or x = 12.â€ť Guide the student to consider both the absolute value of zero and the absolute value of negative numbers in light of the concept of absolute value as distance from zero on a number line. Provide opportunities to solve absolute value equations. Consider implementing MFAS task Solving Absolute Value Equations (ACED.1.1).
Review simple inequalities, compound inequalities, and graphs of inequalities. Be sure the student understands the distinction between inclusive and exclusive inequality symbols and conventions associated with graphing inequalities (e.g., closed versus open dots) and using interval notation (e.g., brackets versus parentheses). Then introduce absolute value inequalities. 
Moving Forward 
Misconception/Error The student does not understand how to rewrite absolute value inequalities as equivalent compound inequalities without absolute value symbols. 
Examples of Student Work at this Level When attempting to rewrite absolute value inequalities without absolute value symbols, the student:
 Uses the wrong inequality symbol or no inequality symbol.
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 Rewrites each inequality as an equation.
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 Changes subtractions within the absolute value symbols to additions.
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 Writes each solution as a disjunction.
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 Writes each solution as a conjunction.
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Questions Eliciting Thinking If the absolute value of some number is less than 4.7, what are some possible values of that number?
If the absolute value of some number is greater than or equal to 12.5, what are some possible values of that number?
Can a number be both greater than or equal to 12.5 and less than or equal to 12.5? 
Instructional Implications Use number lines to illustrate solution sets of simple absolute value inequalities such as x < 4 or x = 6. Relate the meaning of these kinds of inequalities to the studentâ€™s understanding of absolute value as a numberâ€™s distance from zero. For example, explain that x < 4 describes all numbers, x, whose distance from zero on the number line is less than four. Refer to the graph of x < 4 as an illustration of this set of numbers. Then guide the student to represent this set of numbers using a compound inequality such as 4 < x < 4 (x < 4 and x > 4) or interval notation such as (4, 4). Provide additional examples of absolute value inequalities using the greater than (or greater than or equal to) symbols. Emphasize the distinction between conjunctions and disjunctions.
Guide the student to be precise in the use of the words and and or. Make clear that two conditions joined by the word and must both be met. For example, if x < 4 and x > 4 then x must be a number that meets both the conditions that it is less than 4 and greater than 4 to be a part of the solution set. Explain that a solution given using the word or such as x = 6 or x = 6 describes a set of numbers that is either less than 6 or greater than six. Be very clear that a number cannot meet both of these conditions but must meet one or the other to be a part of the solution set. In either case, ask the student to generate examples of solutions. Relate the sample solutions to graphs of the solution sets.
Be sure the student understands the distinction between inclusive and exclusive inequality symbols and conventions associated with graphing inequalities (e.g., closed versus open dots) and using interval notation (e.g., brackets versus parentheses). 
Almost There 
Misconception/Error The student makes a computational or sign error when solving an inequality. 
Examples of Student Work at this Level The student correctly rewrites each absolute value inequality as an equivalent compound inequality without absolute value symbols. However, the student makes a computational error or drops a negative symbol in the process of solving. 
Questions Eliciting Thinking There is mistake in your solution for this inequality. Can you find it?
Did you check to see if any values from your solution set satisfy the original inequality? 
Instructional Implications Provide feedback to the student concerning any errors made. If needed, review solving multistep equations and inequalities. Provide additional opportunities to solve multistep absolute value inequalities. 
Got It 
Misconception/Error The student provides complete and correct responses to all components of the task. 
Examples of Student Work at this Level The student correctly solves each inequality and represents the solution sets using appropriate notation:
 4.7< x < 4.7 ( x < 4.7 and x > 4.7 )
 xÂ = 12.5 or x = 12.5
 4.5= x = 1.5 ( x = 1.5 and x = 4.5 )
 xÂ > 5 or x < 2.5

Questions Eliciting Thinking How did you know to use the word and when writing the solution of the first inequality? What are some examples of solutions of the first inequality? Is 4.7 a solution?
How did you know to use the word or when writing the solution of the second inequality? What are some examples of solutions of the second inequality? Is 12.5 a solution?
Does this inequality have any solutions: x = 4? 
Instructional Implications Introduce the student to writing absolute value inequalities to solve problems. Provide opportunities to solve word problems by writing and solving absolute value inequalities.
Consider implementing MFAS task Writing Absolute Value Inequalities (ACED.1.1). 