Mathematics
Grade 9
15 min
Subtraction word problems - up to two digits
Subtraction word problems - up to two digits
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Introduction & Learning Objectives
Learning Objectives
Translate verbal phrases involving subtraction and limits (e.g., 'decreased by', 'at least', 'no more than') into one-variable linear inequalities.
Model real-world scenarios involving differences or decreases using variables and appropriate inequality symbols.
Solve single-variable linear inequalities that arise from subtraction word problems, including those with negative coefficients.
Represent the solution sets of these inequalities on a number line, using open and closed circles correctly.
Interpret the solution set in the context of the original word problem, considering practical constraints on the variable.
Set up and solve compound inequalities derived from word problems about a range or difference.
You have a $75 gi...
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Key Concepts & Vocabulary
TermDefinitionExample
Linear InequalityA mathematical statement that compares two expressions using an inequality symbol (<, >, ≤, ≥), where the variable is raised to the first power.The cost `c` must be less than $50. This is written as `c < 50`. A problem might state: 'After a $15 discount, the price is less than $50', which models as `p - 15 < 50`.
Solution SetThe set of all numbers that, when substituted for the variable, make the inequality a true statement.For the inequality `x - 10 > 25`, the solution is `x > 35`. The solution set includes 36, 40.5, 100, and infinitely many other numbers greater than 35.
Translating Subtraction PhrasesConverting words that imply subtraction into algebraic expressions.'A number `n` decreased by 21' translates to...
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Core Formulas
Addition/Subtraction Property of Inequality
If `a < b`, then `a + c < b + c` and `a - c < b - c`.
Use this rule to isolate the variable by adding or subtracting a constant from both sides of the inequality. The direction of the inequality symbol does not change.
Multiplication/Division Property of Inequality (Negative Number)
If `a < b` and `c < 0`, then `ac > bc` and `a/c > b/c`.
This is a critical rule. When you multiply or divide both sides of an inequality by a negative number, you MUST reverse the direction of the inequality symbol. This often occurs in problems like `50 - x > 20`.
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Challenging
A scientist has a 95 mL solution. She needs to remove a certain amount, 'x' mL, for an experiment. The remaining volume must be at least 40 mL to preserve for a second experiment, but for safety, the amount removed, 'x', cannot be more than 30 mL. Which inequality represents the valid amount 'x' that can be removed?
A.0 < x ≤ 30
B.30 ≤ x ≤ 55
C.0 < x ≤ 55
D.40 ≤ x ≤ 95
Challenging
A car's fuel tank has a capacity of 60 liters. The car starts with a full tank. After driving for some time, 'L' liters of fuel have been used. The fuel gauge is faulty and only shows 'low fuel' when the remaining fuel is less than 10 liters. However, the driver wants to refuel when there are at least 15 liters left. Which inequality represents the amount of fuel used, 'L', for which the driver wants to refuel but the 'low fuel' light is NOT on?
A.L > 50
B.45 ≤ L < 50
C.L ≤ 45
D.10 < L ≤ 15
Challenging
An initial value of 70 is reduced by an amount 'x', which must be a multiple of 5. The final result must be greater than 20 but no more than 40. What is the sum of all possible values of x?
A.105
B.110
C.140
D.150
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