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Learn to graph systems of inequalities in two variables from a graphical perspective. Understand how to determine solution sets using graphical representations and test points effectively. Practice graphing inequalities step by step.
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College Algebra Fifth Edition James StewartLothar RedlinSaleem Watson
Systems of Inequalities • In this section, we study: • Systems of inequalities in two variables from a graphical point of view.
Graphing an Inequality • We begin by considering the graph of a single inequality.
Graphing an Inequality • We already know that the graph of y =x2, for example, is the parabolashown. • If we replace the equal sign by the symbol ≥, we obtain the inequality y ≥x2
Graphing an Inequality • Its graph consists of not just the parabola shown, but also every point whose y-coordinate is largerthan x2.
Graphing an Inequality • We indicate the solution here by shading the points abovethe parabola.
Graphing an Inequality • Similarly, the graph of y ≤x2 consists of all points on and belowthe parabola.
Graphing an Inequality • However, the graphs of y > x2 and y < x2 do not include the points on the parabola itself, as indicated by the dashed curves here.
Graphing an Inequality • The graph of an inequality, in general, consists of: • A region in the plane whose boundary is the graph of the equation obtained by replacing the inequality sign (≥, ≤, >, or <) with an equal sign.
Graphing an Inequality • To determine which side of the graph gives the solution set of the inequality, we need only check test points.
Graphing Inequalities • To graph an inequality, we carry out these steps. • Graph equation. • Test points.
Graphing Inequalities—Step 1 • Graph equation. • Graph the equation corresponding to the inequality. • Use a dashed curve for > or <, and a solid curve for ≤ or ≥.
Graphing Inequalities—Step 2 • Test points. • Test one point in each region formed by the graph in step 1. • If the point satisfies the inequality, then all the points in that region satisfy the inequality. • In that case, shade the region to indicate it is part of the graph. • If the test point does not satisfy the inequality, then the region isn’t part of the graph.
E.g. 1—Graphs of Inequalities • Graph each inequality. • (a) x2 + y2 < 25 • (b) x + 2y ≥ 5
Example (a) E.g. 1—Graphs of Inequalities • The graph of x2 + y2 = 25 is a circle of radius 5 centered at the origin. • The points on the circle itself do not satisfy the inequality because it is of the form <. • So, we graph the circle with a dashed curve.
Example (a) E.g. 1—Graphs of Inequalities • To determine whether the inside or the outside of the circle satisfies the inequality, we use the test points: • (0, 0) on the inside. • (6, 0) on the outside.
Example (a) E.g. 1—Graphs of Inequalities • To do this, we substitute the coordinates of each point into the inequality and check if the result satisfies the inequality. • Note that anypoint inside or outside the circle can serve as a test point. • We have chosen these points for simplicity.
Example (a) E.g. 1—Graphs of Inequalities • Thus, the graph of x2 + y2 < 25 is the set of all points insidethe circle.
Example (b) E.g. 1—Graphs of Inequalities • The graph of x + 2y = 5 is the line shown. • We use the test points (0, 0) and (5, 5) on opposite sides of the line.
Example (b) E.g. 1—Graphs of Inequalities • Our check shows that the points abovethe line satisfy the inequality.
Example (b) E.g. 1—Graphs of Inequalities • Alternatively, we could put the inequality into slope-intercept form and graph it directly: • x + 2y ≥ 5 • 2y ≥ –x + 5 • y ≥ –1/2x + (5/2)
Example (b) E.g. 1—Graphs of Inequalities • From this form, we see the graph includes all points whose y-coordinates are greaterthan those on the line y = –1/2x + (5/2). • That is, the graph consists of the points on or abovethis line, as shown.
Systems of Inequalities • The solution of a system of inequalities is: • The set of all points in the coordinate plane that satisfy every inequality in the system.
E.g. 2—System of Two Inequalities • Graph the solution of the system of inequalities. • These are the two inequalities of Example 1. • Here, we wish to graph only those points that simultaneously satisfy both inequalities. • The solution consists of the intersection of the graphs in Example 1.
E.g. 2—System of Two Inequalities • In (a), we show the two regions on the same coordinate plane (in different colors). • In (b), we show their intersection.
E.g. 2—System of Two Inequalities: Vertices • The points (–3, 4) and (5, 0) in (b) are the vertices of the solution set. • They are obtained by solving the system of equations • We do so by substitution.
E.g. 2—System of Two Inequalities: Vertices • Solving for x in the second equation gives: x = 5 – 2y • Substituting this into the first equation gives: • (5 – 2y)2 + y2 = 25 • (25 – 20y + 4y2) + y2 = 25 • –20y + 5y2 = 0 • –5y(4 – y) = 0
E.g. 2—System of Two Inequalities: Vertices • Thus, y = 0 or y = 4. • When y = 0, we have: x = 5 – 2(0) = 5 • When y = 4, we have: x = 5 – 2(4) = –3 • So, the points of intersection of these curves are (5, 0) and (–3, 4).
E.g. 2—System of Two Inequalities: Vertices • Note that, in this case, the vertices are not part of the solution set. • This is because they don’t satisfy the inequality x2 + y2 < 25. • So, they are graphed as open circles in the figure. • They simply show where the “corners” of the solution set lie.
Systems of Linear Inequalities • An inequality is linearif it can be put into one of these forms:ax + by ≥c ax + by ≤c ax + by >c ax + by <c
E.g. 3—System of Four Linear Inequalities • Graph the solution set of the system, and label its vertices.
E.g. 3—System of Four Linear Inequalities • We first graph the lines given by the equations that correspond to each inequality.
E.g. 3—System of Four Linear Inequalities • For the graphs of the linear inequalities, we only need to check one test point. • For simplicity, let’s use the point (0, 0).
E.g. 3—System of Four Linear Inequalities • Since (0, 0) is below the line x + 3y = 12, our check shows that the region on or below the line must satisfy the inequality. • Likewise, since (0, 0) is below the line x + y = 8, our check shows that the region on or below this line must satisfy the inequality.
E.g. 3—System of Four Linear Inequalities • The inequalities x ≥ 0 and y ≥ 0 say that x and y are nonnegative. • These regions are sketched in (a). • The intersection, the solution set, is shown in (b).
E.g. 3—System of Four Linear Inequalities: Vertices • The coordinates of each vertex are obtained by simultaneously solving the equations of: • The lines that intersect at that vertex.
E.g. 3—System of Four Linear Inequalities: Vertices • From the system we get the vertex (6, 2).
E.g. 3—A System of Four Linear Inequalities: Vertices • The other vertices are the x- and y-intercepts of the corresponding lines, (8, 0) and (0, 4), and the origin (0, 0). • Here, all the vertices arepart of the solution set.
E.g. 4—System of Linear Inequalities • Graph the solution set of the system. • We must graph the lines that correspond to these inequalities and then shade the appropriate regions—as in Example 3.
E.g. 4—System of Linear Inequalities • We will use a graphing calculator. • So, we must first isolate y on the left-hand side of each inequality.
E.g. 4—System of Linear Inequalities • Using the shading feature of the calculator, we obtain this graph. • The solution set is the triangular region that is shaded in all three patterns. • We then use TRACE or the Intersect command to find the vertices of the region.
E.g. 4—System of Linear Inequalities • The solution set is graphed here.
Bounded Regions • When a region in the plane can be covered by a (sufficiently large) circle, it is said to be bounded.
Unbounded Regions • A region that is not bounded is called unbounded. • It cannot be “fenced in”—it extends infinitely far in at least one direction.
