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Understanding Linear Inequalities and Their Graphs

This guide covers the key concepts of solving and graphing linear inequalities in two variables. Students will learn to distinguish between different types of solutions, such as consistent, inconsistent, and dependent systems. Along with step-by-step methods for graphing these inequalities, this resource includes practice problems and a benchmark test to reinforce learning. By the end of the lessons, students will confidently solve systems of linear inequalities and interpret their graphs in coordinate planes.

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Understanding Linear Inequalities and Their Graphs

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  1. Do Now 1/20/10 • Take out HW from last night. • Text p. 462, #1-8 all, #10, #12, #16-30 evens, #36 • Copy HW in your planner. • Benchmark Test #1 evens • Text p. 469, #3-8 all, #10-38 evens • Quiz sections 7.5 – 7.6 Friday

  2. 1) inconsistent 2) consistent dependent 3) lines have same slope but different y-intercepts 4) the graph would show only one line 5) B; one solution 6) C; no solution 7) A; infinitely many solutions 8) no solution 10) one solution 12) infinitely many solutions 16) infinitely many solutions 18) infinitely many solutions 20) no solution 22) (3,0) 24) C 26) no solution 28) one solution 30) one solution 36) No, there are infinitely many solutions Homework Text p. 462, #1-8 all, #10, #12, #16-30 evens & 36

  3. Objective • SWBAT solve systems of linear inequalities in two variables

  4. Section 6.7 “Graph Linear Inequalities” Remember This??? Linear Inequalities- the result of replacing the = sign in a linear equation with an inequality sign. 2x + 3y > 4 y ≥ 4x - 3 y ≤ ½x + 3 7y < 8x - 16

  5. Remember This??? Graphing Linear Inequalities • Graphing Boundary Lines: • Use a dashed line for < or >. • Use a solid line for ≤ or ≥.

  6. Graph an Inequality Remember This??? Graph the inequality y > 4x - 3. STEP2 STEP3 STEP1 Graph the equation Test (0,0) in the original inequality. Shade the half-plane that contains the point (0,0), because (0,0) is a solution to the inequality.

  7. Graph an Inequality Remember This??? Graph the inequality x + 3y ≥ -1. STEP2 STEP3 STEP1 Shade the half-plane that contains the point (1,0), because (1,0) is a solution to the inequality. Graph the equation Test (1,0) in the original inequality.

  8. Graph an Inequality Remember This??? Graph the inequality y ≥ -3. STEP2 STEP3 STEP1 Shade the half-plane that contains the point (2,0), because (2,0) is a solution to the inequality. Graph the equation Test (2,0) in the original inequality. Use only the y-coordinate, because the inequality does not have a x-variable.

  9. Graph an Inequality Remember This??? Graph the inequality x≤ -1. STEP2 STEP3 STEP1 Shade the half-plane that does not contain the point (3,0), because (3,0) is not a solution to the inequality. Graph the equation Test (3,0) in the original inequality. Use only the y-coordinate, because the inequality does not have a x-variable.

  10. Section 7.6 “Solve Systems of Linear Inequalities” SYSTEM OF INEQUALITIES- consists of two or more linear inequalities in the same variables. x – y > 7 Inequality 1 2x + y < 8 Inequality 2 A solution to a system of inequalities is an ordered pair (a point) that is a solution to both linear inequalities.

  11. ? ? 1 > 0 – 2 1 > 0 + 6 1 > – 2 1 > 6 Graph a System of Inequalities y > -x – 2 Inequality 1 y ≤ 3x + 6 Inequality 2 Graph both inequalities in the same coordinate plane. The graph of the system is the intersection of the two half-planes, which is shown as the darker shade of blue. (0,1)

  12. ? ? 0 < 5 – 4 0≥-5 + 3 0 < 1 0≥ -2 Graph a System of Inequalities y < x – 4 Inequality 1 y ≥ -x + 3 Inequality 2 Graph both inequalities in the same coordinate plane. The graph of the system is the intersection of the two half-planes, which is shown as the darker shade of blue. (5,0)

  13. ? ? ? x + 2y ≤ 4 y ≥ -1 x > -2 0≥ -1 0 + 0 ≤ 4 0 > -2 Graph a System of THREE Inequalities y ≥ -1 Inequality 1: Graph all three inequalities in the same coordinate plane. The graph of the system is the triangular region, which is shown as the darker shade of blue. x > -2 Inequality 2: x + 2y ≤ 4 Inequality 3: Check (0,0)

  14. Graph a System of THREE Inequalities y ≥ -x + 2 y > -x Inequality 1: Inequality 1: y < 4 y ≥ x – 4 Inequality 2: Inequality 2: x < 3 y < 5 Inequality 3: Inequality 3:

  15. Write a system of inequalities for the shaded region. y > 2x + 1 y ≥ 3 Inequality 1 INEQUALITY 1: One boundary line for the shaded region is y = 3. Because the shaded region is above the solid line, the inequality is y ≥ 3. Inequality 2 Write a System of Linear Inequalities INEQUALITY 2: Another boundary line for the shaded region has a slope of 2 and a y-intercept of 1. So, its equation is y = 2x + 1. Because the shaded region is above the dashed line, the inequality is y > 2x + 1.

  16. Homework NJASK7 Prep • Text p. 469, #3-8 all, #10-38 evens

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