Linear Inequalities in Two Variables and Systems of Inequalities

4.6: Linear Inequalities in Two Variables and Systems of Inequalities. Original editorial SAT preparation with three worked examples, ten flashcards, and thirty practice questions. Educator-review draft.

Lesson 6 of 63 parts Practice Flashcards

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Test an ordered pair

Objective: Determine whether a point satisfies a linear inequality.

Prerequisite skills: coordinate substitution and solving inequalities. Substitute both coordinates into the inequality. A true numerical comparison means the point belongs to the solution half-plane. The same test locates a shading side without relying on an ambiguous drawing. Common error: testing only the x-coordinate or automatically including the boundary. Restrictions and interpretation: A boundary line is included for ≤ or ≥ and excluded for < or >; a system requires all conditions.

Formulas

$(a,b) satisfies y > mx + c iff b > ma + c$

Worked examples

Try itWhich point satisfies y > 2x + 21?Show answer

For (1, 24), the boundary output is 2 + 21 = 23; its y-value is one larger, so the strict inequality holds. The point (1, 23) is on the excluded boundary, and both remaining points lie below it. Therefore the requested answer is (1, 24).

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Test an ordered pair

Objective: Determine whether a point satisfies a linear inequality.

Formulas

$(a,b) satisfies y > mx + c iff b > ma + c$

Worked examples

Try itWhich point satisfies y > 2x + 21?Show answer

Intersect two half-planes

Objective: Require a point to satisfy every inequality in a system.

The solution of a system of inequalities is the overlap of their half-planes. A point passing only one condition is not a system solution. Include any nonnegativity or integrality restrictions separately. Common error: accepting a point that meets only one of the conditions. Restrictions and interpretation: A boundary line is included for ≤ or ≥ and excluded for < or >; a system requires all conditions.

Formulas

solution region = intersection of individual solution sets

Worked examples

Try itWhich point satisfies both y ≥ x + 22 and y ≤ 27?Show answer

At (2, 25), the lower bound is 24 and the upper bound is 27, so both comparisons hold. The other points violate the lower bound, upper bound, and lower bound respectively. Therefore the requested answer is (2, 25).

Maximum feasible integer value

Objective: Combine linear inequalities and discrete restrictions.

When one variable is fixed, each inequality yields a restriction on the other. Intersect those restrictions before choosing an integer extremum. A rounded graph can hide whether a boundary is included. Common error: optimizing using only one constraint. Restrictions and interpretation: A boundary line is included for ≤ or ≥ and excluded for < or >; a system requires all conditions. 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

$ax + by ≤ C ⇒ y ≤ (C − ax)/b when b > 0$

Worked examples

Try itNonnegative integers x and y satisfy 3x + 4y ≤ 189 and y ≤ 31. If x = 25, what is the greatest possible y?Show answer

Substitute x = 25: 4y ≤ 114, so y ≤ 28.5. The second bound y ≤ 31 is looser. With integer y, the maximum is 28. Therefore the requested answer is 28.

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