Linear Inequalities in One Variable

4.5: Linear Inequalities in One Variable. Original editorial SAT preparation with three worked examples, ten flashcards, and thirty practice questions. Educator-review draft.

Lesson 5 of 63 parts Practice Flashcards

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Positive coefficient inequalities

Objective: Solve and interpret a one-variable inequality.

Prerequisite skills: solving linear equations and comparing real numbers. Adding or subtracting the same quantity preserves inequality direction. Multiplying or dividing by a positive number also preserves it. An open endpoint excludes equality; a closed endpoint includes it. Common error: including an endpoint for a strict inequality. Restrictions and interpretation: Reverse an inequality when multiplying or dividing by a negative number.

Formulas

$ax + b ≤ c ⇒ x ≤ (c − b)/a for a > 0$

Worked examples

Try itWhich describes all x satisfying 3x + 5 < 74?Show answer

Subtract 5 to get 3x < 69. Divide by positive 3: x < 23. The endpoint is excluded because the original inequality is strict. Therefore the requested answer is x < 23.

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Positive coefficient inequalities

Objective: Solve and interpret a one-variable inequality.

Formulas

$ax + b ≤ c ⇒ x ≤ (c − b)/a for a > 0$

Worked examples

Try itWhich describes all x satisfying 3x + 5 < 74?Show answer

Subtract 5 to get 3x < 69. Divide by positive 3: x < 23. The endpoint is excluded because the original inequality is strict. Therefore the requested answer is x < 23.

Negative coefficient inequalities

Objective: Reverse direction after division by a negative.

Multiplication by a negative reverses order: if a < b, then −a > −b. The same principle applies to every term of an inequality. Check one value on each side of the boundary if uncertain. Common error: keeping the arrow direction after dividing by a negative. Restrictions and interpretation: Reverse an inequality when multiplying or dividing by a negative number.

Formulas

$−ax < b ⇒ x > −b/a for a > 0$

Worked examples

Try itWhich describes all x satisfying −4x + 7 ≥ -85?Show answer

Subtract 7: −4x ≥ -92. Divide by −4 and reverse the inequality: x ≤ 23. Equality is included. Therefore the requested answer is x ≤ 23.

Integer constraints

Objective: Use an inequality to find a feasible integer maximum.

After solving a budget or capacity inequality, apply the input restrictions. If only whole items can be bought, take the greatest integer not exceeding an upper bound. Rounding to the nearest integer can exceed the budget. Common error: rounding an upper bound upward and violating the original constraint. Restrictions and interpretation: Reverse an inequality when multiplying or dividing by a negative number. Desmos and method choice: Use Desmos for arithmetic or a graph/table check when that reduces work. Enter parentheses explicitly and verify the requested quantity and units. Use the algebraic steps shown here for an exact result; a rounded graph display is not proof of exact equality.

Formulas

x ≤ B/p with integer x ⇒ maximum x = floor(B/p)

Worked examples

Try itA club has $212. It pays a $13 fee and buys kits costing $7 each. What is the maximum number of whole kits it can buy?Show answer

Use 13 + 7k ≤ 212. Then k ≤ 199/7. Whole kits require k ≤ 28. Buying 29 kits would cost $216, exceeding the budget. Therefore the requested answer is 28.

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