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Graphs of Linear Inequalities

3 x – y < 5. 3 x – y < 5. 3( 1 ) – 5 5. 3( 6 ) – ( –2 ) 5. 20 < 5. FALSE. –2 < 5. TRUE. Graphs of Linear Inequalities.

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Graphs of Linear Inequalities

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  1. 3x – y < 5 3x – y < 5 3(1)–5 5 3(6) – (–2) 5 20 < 5 FALSE –2 < 5 TRUE Graphs of Linear Inequalities When the equal sign in a linear equation is replaced with an inequality sign, a linear inequality is formed. Solutions of linear inequalities are ordered pairs. Example Determine whether (1, 5) and (6, –2) are solutions of the inequality 3x – y < 5. Solution The pair (1, 5) is a solution of the inequality, but (6, –2) is not.

  2. 3x – y < 5 3x – y < 5 3(1)–5 5 3(6) – (–2) 5 20 < 5 FALSE –2 < 5 TRUE Example Determine whether (1, 5) and (6, –2) are solutions of the inequality 3x – y < 5. Solution The pair (1, 5) is a solution of the inequality, but (6, –2) is not.

  3. Graph Example Solution First: Graph the boundary y = x. Since the inequality is greater than or equal to, the line is drawn solid and is part of the graph of (0, 1) The graph of a linear equation is a straight line. The graph of a linear inequality is a half-plane, with a boundary that is a straight line. To find the equation of the boundary line, we simply replace the inequality sign with an equals sign. y 6 y = x 5 4 3 2 1 -5 -4 -3 -2 -1 1 2 3 4 5 x -1 -2 -3 -4 -5 Second: We choose a test point on one side of the boundary, say (0, 1). Substituting into the inequality we get We finish drawing the solution set by shading the half-plane that includes (0, 1).

  4. Examples Graph the following inequalities.

  5. Systems of Linear Inequalities To graph a system of equations, we graph the individual equations and then find the intersection of the individual graphs. We do the same thing for a system of inequalities, that is, we graph each inequality and find the intersection of the individual graphs.

  6. The graph of the system Example

  7. Type Example Solution Linear inequalities –3x + 5 > 2 A set of numbers; in one variable an interval Type Example Solution Linear equations 2x – 8 = 3(x + 5) A number in one variable Graph Graph Let’s look at 6 different types of problems that we have solved, along with illustrations of each type.

  8. Type Example Solution Linear equations 2x + y = 7 A set of ordered in two variables pairs; a line Graph

  9. Type Example Solution Linear inequalities x + y≥ 4 A set of ordered in two variables pairs; a half-plane Graph

  10. Type Example Solution System of x + y= 3, An ordered pair or equations in 5x – y = –27 a (possibly empty) two variables set of ordered pairs Graph

  11. Type Example Solution System of 6x – 2y ≤ 12, A set of ordered inequalities in y – 3 ≤ 0, pairs; a region two variables y ≥ x of a plane Graph

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