Solving and Graphing Quadratic Inequalities

7 Key excerpts on "Solving and Graphing Quadratic Inequalities"

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  eBook - ePub

  Algebra II For Dummies

  • Mary Jane Sterling(Author)
  • 2018(Publication Date)
  • For Dummies

    (Publisher)

  y is bigger than or smaller than. Often, the graph of a system gives you more information than the listing of the inequalities shown in an algebraic solution. You can see that all the points in the solution lie above a certain line, so you pick numbers that work in the system based on what you see.

  Drawing and quartering inequalities

  The simplest inequality to graph and solve is one that falls above or below a horizontal line or to the right or left of a vertical line. A system of inequalities that involves two such lines (one vertical and one horizontal) has a graph that appears as one quarter of the plane. For instance, the graph of the system of inequalities is shown in Figure 13-14 .

  John Wiley & Sons, Inc.

  FIGURE 13-14: Two inequalities intersecting to share a portion of the plane (the heavy shading).

  Everything to the right of the line represents the graph of , and everything below the line represents the graph of . Their intersection, the heavily shaded area in the lower-right quadrant formed by the intersecting lines, consists of all the points in that shaded area. You can find an infinite number of solutions. Some examples are the points (3, 1), (4, 2), and . You could never list all the answers.

  Graphing areas with curves and lines

  You find the solution of a system of inequalities involving a line and a curve (such as a parabola), two curves, or any other such combination by graphing the individual equations, determining which side to shade for each curve, and identifying where the equations share the shading.

  To solve the system , for example, make your way through the following steps:

  1. Graph the line and determine which side of the line to shade by checking a test point (a random point that’s clearly on one side or the other) to see if it satisfies the inequality. (For info on graphing lines, see

     Chapters 2

     and

     5

     .)
     If the point satisfies the inequality, you shade that side. In this case, you can use the test point (0, 0), which is clearly above and to the left of the line. When you put the coordinate (0, 0) in the inequality , you get . Yes, 0 is greater than

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  Intermediate Algebra

  A Guided Approach

  • Rosemary Karr, Marilyn Massey, R. Gustafson, , Rosemary Karr, Marilyn Massey, R. Gustafson(Authors)
  • 2014(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  The solution set is 1 2` , 2 2 2 < 1 0, 1 2 . Solve 2 x 1 2 . 1 x and graph the solution set. EXAMPLE 4 Solution a SELF CHECK 4 To find the solutions of x 2 1 2 x 2 3 $ 0 (Example 1) by graphing with a TI-84 calculator, we can use window settings of 3 2 10, 10 4 for x and 3 2 10, 10 4 for y and graph the qua-dratic function y 5 x 2 1 2 x 2 3 , as in Figure 8-26 on the next page. The solutions of the inequality will be those values of x for which the graph of y 5 x 2 1 2 x 2 3 lies above or on the x -axis. We can use the zero feature to find the critical values and interpret the graph to determine that the solution is the interval 1 2` , 2 3 4 < 3 1, ` 2 . Accent on technology  Solving Inequalities (continued) Unless otherwise noted, all content on this page is © Cengage Learning. Copyright 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 584 CHAPTER 8 Quadratic Equations and Functions; Inequalities; and Algebra, Composition, and Inverses of Functions y = x 2 + 2 x – 3 Figure 8-26 To find the solutions of 3 x 2 1 , 2 x (Example 4), we first write the inequality in the form 3 x 2 1 2 2 x , 0 Then we use window settings of 3 2 5, 5 4 for x and 3 2 3, 3 4 for y and graph the function y 5 3 x 2 1 2 2 x , as in Figure 8-27(a). The solutions of the inequality will be those numbers x for which the graph lies below the x -axis. We can trace to see that the graph is below the x -axis when x is less than 2 2 . Since we cannot see the graph in the interval 0 , x , 1 , we redraw the graph using window settings of 3 2 1, 2 4 for x and 3 2 25, 10 4 for y .

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  Algebra

  A Combined Course 2E

  • Charles P. McKeague(Author)
  • 2018(Publication Date)
  • XYZ Textbooks

    (Publisher)

  Chapter 12 Quadratic Equations and Functions 990 Equations Quadratic in Form [12.4] There are a variety of equations whose form is quadratic. We solve most of them by making a substitution so the equation becomes quadratic, and then solving the equation by factoring or the quadratic formula. For example, The equation is quadratic in (2x − 3) 2 + 5(2x − 3) − 6 = 0 2x − 3 4x 4 − 7x 2 − 2 = 0 x 2 2x − 7 √ — x + 3 = 0 √ — x Graphing Quadratic Functions [12.5] The graph of any function of the form f (x) = ax 2 + bx + c a ≠ 0 is a parabola. The graph is concave up if a > 0 and concave down if a < 0. The highest or lowest point on the graph is called the vertex and always has an x-coordinate of x = −b ___ 2a . Quadratic Inequalities [12.6] We solve quadratic inequalities by manipulating the inequality to get 0 on the right side and then factoring the left side. We then make a diagram that indicates where the factors are positive and where they are negative. From this sign diagram and the original inequality we graph the appropriate solution set. Rational Inequalities [12.6] We solve rational inequalities by simplifying the rational expressions on the left side and obtaining 0 on the right side. We then make a diagram that indicates where the factors are positive and where they are negative. From this sign diagram and the original inequality we graph the appropriate solution set, making sure to avoid zero denominators. 5. The equation x 4 − x 2 − 12 = 0 is quadratic in x 2 . Letting y = x 2 we have y 2 − y − 12 = 0 ( y − 4)( y + 3) = 0 y = 4 or y = −3 Resubstituting x 2 for y, we have x 2 = 4 or x 2 = −3 x = ±2 x = ±i √ — 3 6. The graph of y = x 2 − 4 will be a parabola. It will cross the x-axis at 2 and −2, and the vertex will be (0, −4). 7. Solve x 2 − 2x − 8 > 0. We factor and draw the sign diagram. (x − 4)(x + 2) > 0 The solution set is (−∞, −2) ∪ (4, ∞). x - 4 -2 4 x + 2 - - - - - - - - - - - - + + + + + + + + + + + + 8.

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  eBook - ePub

  Linear Mathematics

  A Practical Approach

  • Patricia Clark Kenschaft(Author)
  • 2013(Publication Date)
  • Dover Publications

    (Publisher)

  Chapter 5

  INTRODUCTION TO LINEAR PROGRAMMING

  5.1 Graphing Linear Inequalities

  This section discusses how to graph a linear inequality in two variables, because this skill will be needed in the next section. First we review numerical inequalities.

  If you add the same number to both sides of an inequality, the inequality remains true (Figure 5.1–1 ). Similarly, if you multiply both

  Figure 5.1–1

  sides of an inequality by the same positive number , the inequality remains true (Figure 5.1–2 ). On the other hand, if you multiply an inequality by a negative number , the inequality is reversed (Figure 5.1–3 ). These facts are used in manipulating expressions such as the following:

  Figure 5.1–2

  Figure 5.1–3

  When variables appear in an inequality, there will usually be some values of the variable(s) that make it true and others that make it false. When the values that make it true are pictured on a graph, we say we have “graphed” that inequality. You have had some experience in graphing linear equations (if you want more, consult Appendix 2 ), and you may remember that the graph of a linear equation is always a straight line. By contrast, the graph of a linear inequality is a whole region.

  5.1.1 Example

  Graph 2x + y ≥ 4.

  Solution:

  It is helpful to subtract 2x from each side of this inequality and write it in the form

  y ≥ 4 – 2x

  Now consider the equation y = 4 – 2x. It is the straight line indicated in Figure 5.1–4 with y-intercept of 4 and x-intercept of 2. For each x1 , the corresponding y1 such that y1 = 4 – 2x1 will determine the point (x1 , y1 ) lying on the line. If x1 remains the same, but y grows larger than y1 = 4 – 2x1 the point (x1 , y) will be above the line. This is true no matter how much y may grow greater than y1 Thus the points on and above the line y = 4 – 2x form the graph of y ≥ 4 – 2x

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  Precalculus, Enhanced Edition

  • David Cohen, Theodore Lee, David Sklar, , David Cohen, Theodore Lee, David Sklar(Authors)
  • 2016(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  Similarly, for a geometric interpretation of the inequality in Example 1(b), we graph the two lines y 5 t 8 and y 7(1 t ) in a t -y coordinate system. As indi-cated in Figure 3(b), the line y 5 t 8 is below the line y 7(1 t ) provided that t is to the right of 1 2. Furthermore, at t 1 2, the lines intersect. That is, for t 1 2, the quantities 5 t 8 and 7(1 t ) are equal. Algebraically, we can restate this information by saying This confirms the result in Example 1(b). In the next example, we solve the inequality By definition, this is equivalent to the pair of inequalities One way to proceed here would be first to determine the solution set for each inequal-ity. Then the set of real numbers common to both solution sets would be the solution set for the original inequality. However, when we are able to isolate the variable 1 2 3 x 4 and 3 x 4 1 2 1 2 3 x 4 1 2 5 t 8 7(1 t ) provided that t 1 2 2.3 Inequalities 105 Figure 3 (b) The g raph indicates that if t≥1/2 then 5t+8≤7(1+t). 5 0.5 1.0 1.5 10 15 t y y=5t+8 y=7(1+t) (a) To the left of x=4, the line y=2x-3 is below the line y=5. 2 2 3 4 x y 4 6 5 y=2x-3 y=5 In Section 1.5 we saw that there is a close connection between solving equations and graphing. (Example: The roots of the equation x 2 4 0 are the x -intercepts of the graph of y x 2 4.) It’s also true that solving inequalities and graphing are closely related. Consider, for instance, the inequality 2 x 3 5 that we solved in Example 1(a). To interpret this inequality geometrically, we graph the two equations y 2 x 3 and y 5. Then, as indicated in Figure 3(a), the graph of y 2 x 3 is below the line y 5 as long as x is to the left of 4. The previous sentence translates algebraically into This confirms the result in Example 1(a). 2 x 3 5 provided that x 4 Copyright 201 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

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  Intermediate Algebra

  Connecting Concepts through Applications

  • Mark Clark, Cynthia Anfinson(Authors)
  • 2018(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  Graph quadratic inequalities in two variables. 4.7 Graphing from Standard Form Quadratic functions are often written in standard form rather than in vertex form, so it is important that we learn to graph from the standard form. Since all quadratic functions have a vertex and a vertical intercept, these two points will be the starting points for graphing. A quadratic function in standard form f 1 x 2 5 ax 2 1 bx 1 c gives several key pieces of information about the graph. The value of a in the standard form affects the graph in the same way that it affects vertex form. If a is positive, the parabola opens upward. If a is negative, the parabola opens downward. Also a affects whether the graph is wide or narrow. The value c gives the following information about the graph. When we substitute zero into the input variable for the function, the output is c. Let f 1 x 2 5 ax 2 1 bx 1 c and x 5 0 f 1 02 5 a1 02 2 1 b1 02 1 c f 1 02 5 c which means that (0, c) is the vertical intercept for the graph. f 1 x 2 5 4x 2 1 3x 2 12 f 1 02 5 41 02 2 1 31 02 2 12 f 1 02 5 212 1 0, 2122 The vertical intercept Another important relationship for parabolas is between the vertex and the standard form of a quadratic function. We will use the Concept Investigation to look at the relationship between the standard form of a quadratic function and the vertex. Connecting the Concepts Is a the same in both forms? The a-value in the vertex form is the same as the a-value in standard form. If we multiply out the vertex form and simplify into the standard form, the a-value does not change. Example: f 1 x2 5 51 x 2 22 2 1 9 f 1 x2 5 51 x 2 221 x 2 22 1 9 f 1 x2 5 51 x 2 2 4x 1 42 1 9 f 1 x2 5 5x 2 2 20x 1 29 The fact that the a-value is the same in both forms means that it will affect the graph of the parabola in the same ways. Positive a: opens upward Negative a: opens downward | a | , 1: wide parabola | a | .

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  Available until 12 Jan |Learn more

  GRE All the Quant

  Effective Strategies & Practice from 99th Percentile Instructors

  • (Author)
  • 2023(Publication Date)
  • Manhattan Prep

    (Publisher)

  true statement.

  However, while equations on the GRE tend to have only one or two values as solutions, inequalities give a whole range of values as solutions—way too many to list individually.

  Here’s a comparison to illustrate:

  Equation	Inequality
  x + 3 = 8−3      −3x = 5	x + 3 < 8−3      −3x < 5

  In the equation, there’s just one solution: x must be 5. If you tried to substitute any number other than 5 for x into the equation you would get a nonsensical answer. For example, 2 + 3 does not equal 8.

  However, there are many different numbers that would make the inequality true—as long as those values are less than 5. Here are just a few possible values of x for the inequality:

  Is this true?	Check the math: x + 3 < 8 ?	Outcome
  x = 2		True, so 2 is a valid solution.
  x = −7		True, so −7 is a valid solution.
  x = 6		False, so 6 is not a valid solution.

  Because inequalities tend to have many possible solutions, this topic is commonly tested in the Select All That Apply format.

  Check Your Skills

  1. Which of the following values of x are solutions to the inequality x < 10 ?
     Indicate all such values.
     −3
     2.5
     9.999

  Answers can be found on page 189.

  Simplifying Inequalities

  Many of the rules you learned with respect to equations will still apply to inequalities; you’ll learn about the few exceptions in this chapter. The rules of PEMDAS apply, and whatever you do to one side of the inequality you must still do to the other side. The goal of isolating a variable remains the same:

  The inequalities 2x + 6 < 12 and x < 3 provide the same information, but the second inequality is much easier to understand.

  Simplifying via Addition and Subtraction

  With equations, if you add the same number to both sides, the equation is still true. The same principle holds true for inequalities. If you add or subtract the same number from both sides, that inequality remains true:

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