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MathLab with Mr. Ahmed is a math learning website by Ahmed Amasha, a mathematics teacher teaching the American curriculum. Explore Algebra 2, Precalculus, Calculus, and SAT and PSAT Math Prep through lessons, focused practice and printable resources.

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10.9 Systems of Inequalities

Focus on system of inequalities and boundary type.

Before you solve

Ideas in this lesson

Explore the lesson brief and the worked example for Idea 11. Create a free account to continue with the remaining ideas and practice.

01Graphing an inequality in two variablesWorked steps · five questions02Systems of inequalitiesWorked steps · five questions
Lesson brief and rules

10.9 Systems of Inequalities

A point solves a system only when it satisfies every inequality.

Worked example 11
01 / Does (4,7)(4,7) satisfy x≥3x\ge 3 and y≥5y\ge 5?
02 / 4≥34\ge 3 and 7≥57\ge 5, so both inequalities hold.
03 / Answer: Yes
Worked example 2 · Boundary type

For x≤4x\le 4, is x=4x=4 included?

The ≤\le sign includes equality. Answer: Yes

Read the graphs

Graphing an inequality in two variables

Graph y>2x−1y > 2x - 1.
Graph for Graphing an inequality in two variablesThe graph corresponds to the stated function or supplied points.−4−224−10−5510xy
Solid: curve 1 · Shading shows y > 2x − 1; the dashed boundary is excluded.

Answer: Dashed line y=2x−1y = 2x - 1, shade above

Systems of inequalities

Graph the system x+y≤5,x≥0,y≥0x + y \le 5, x \ge 0, y \ge 0 and find the vertices.
Graph for Systems of inequalitiesThe graph corresponds to the stated function or supplied points.−1123456−1123456xy
Solid: curve 1 · Shading shows the intersection of all three inequalities.

Answer: Triangle; vertices (0,0),(5,0),(0,5)(0, 0), (5, 0), (0, 5)

15 questions covering every idea, with a worked answer for each.Start practice ↗
Textbook reference: Stewart, Redlin & Watson, Precalculus: Mathematics for Calculus, 7th ed. (Cengage, 2016). The explanations and practice here are original, organized by textbook section.

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