Linear and quadratic inequalities
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Linear and quadratic inequalities. Chapter 9. 9.1 – linear inequalities in two variables. Chapter 9. Linear inequalities. We’ll be looking at linear inequalities in the following form where A, B, and C are real numbers:. Ax + By < C Ax + By ≤ C Ax + By > C Ax + By ≥ C.

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Linear and quadratic inequalities

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Linear and quadratic inequalities

Linear and quadratic inequalities

Chapter 9


9 1 linear inequalities in two variables

9.1 – linear inequalities in two variables

Chapter 9


Linear inequalities

Linear inequalities

We’ll be looking at linear inequalities in the following form where A, B, and C are real numbers:

Ax + By < C

Ax + By ≤ C

Ax + By > C

Ax + By ≥ C

The set of points that satisfy a linear inequality can be called the solution set or the solution region.


Review inequality number lines

Review: Inequality number lines


Boundaries

boundaries

  • A boundary is a line or curve that separates the Cartesian plane into two regions.

  • It may or may not be part of the solution region.

  • Drawn as a solid line and included in the solution region if the inequality involves ≤ or ≥.

  • Drawn as a dashed line and not included in the solution region if the inequality involves < or >.


Example

example

a) Graph 2x + 3y ≤ 6.

b) Determine if the point (–2, 4) is part of the solution.

Solve for y.

  • b) Try putting the point into the inequality and seeing if it holds true.

  • 2x + 3y ≤ 6

  • 2(–2) + 3(4) ≤ 6

  • –4 + 12 ≤ 6

  • 8 ≤ 6

  • This is a false statement, so (–2, 4) is not part of the solution region.

Try graphing 10x – 5x > 0.


Example1

example

Write an inequality to represent the graph.

  • Recall slope-intercept form:

  • y = mx + b

  • b is y-intercept

  • b = 1

  • m is the slope

  • How do you calculate slope?

  • Rise over run!

  • Choose two points.

  • (0, 1) and (1, 3)

  • Rise: 3 – 1 = 2

  • Run: 1 – 0 = 1

  • So: Rise/Run = 2/1 = 2

  • m = 2

  • Is the graph less than or greater than?

  • Is it greater than or greater than or equal to?

y > 2x + 1


Try it

Try it


Example2

example

Suppose that you are constructing a tabletop using aluminum and glass. The most that you can spend on materials is $50. Laminated safety glass costs $60/m2, and aluminum costs $1.75/ft. You can choose the dimensions of the table and the amount of each material used. Find all possible combinations of materials sufficient to make the tabletop.


Pg 472 475 2 4 9 11 12

Pg. 472-475, # 2, 4, 9, 11, 12.

Independent Practice


9 2 quadratic inequalities in one variable

9.2 – quadratic inequalities in one variable

Chapter 9


Example3

example

Solve x2 – 2x – 3 ≤ 0.

Method One:

Graph the function:

For which values of x is the function below the x-axis (below 0 on the y-axis)?

The solution set is all real values of x between –1 and 3, inclusive.

 {x | –1 ≤ x ≤ 3, x E R}


Example4

example

Solve x2 – 2x – 3 ≤ 0.

  • The x-axis is being divided into three intervals. We need to test points from each interval to see which works.

  • Interval One: Try x = –2

  •  (–2)2 – 2(–2) – 3 ≤ 0

  • 5 ≤ 0

  • Doesn’t work

  • Interval Two: Try x = 0

  •  02 – 2(0) – 3 ≤ 0

  • –3 ≤ 0

  • Works!

  • Interval Three: Try x = 4

  • (4)2 – 2(4) – 3 ≤ 0

  • 5 ≤ 0

  • Doesn’t work

Method Two:

Consider x2 – 2x – 3 = 0. Factor it.

(x – 3)(x + 1) = 0

The roots are 3 and –1.

Put these values on a number line, with closed circles since it’s “less than or equal to.”

{x | –1 ≤ x ≤ 3, x E R}


Example5

example

Solve –x2 + x + 12 < 0.

Case 1:

-3

4

Method 3: Case Analysis

  • Factor:

  • –(x2 – x – 12) < 0

  • –(x – 4)(x + 3) < 0

  • Either both factors need to positive, or they are both negative.

  • Case 1: x – 4 < 0 and x + 3 < 0

  • x < 4 and x < –3

  • Case 2: x – 4 > 0 and x + 3 > 0

  •  x > 4 and x > –3

-3

4

Case 2:

-3

4

-3

4

The solution set is values for x for each case that satisfy both inequalities. So, here it is where x < –3 or x > 4.

{x | x < –3 or x > 4, x E R}


Example6

example

Solve 2x2 – 7x > 12.

Rewrite the inequality: 2x2 – 7x – 12 > 0

Find the roots using the

quadratic formula:

Consider the graph:


Example7

example

If a baseball is thrown at an initial speed of 15 m/s from a height of 2 m above the ground, the inequality –4.9t2 + 15t + 2 >0 models the time, t, in seconds, that the baseball is in flight. During what time interval is the baseball in flight?


Pg 484 487 1 4 6 7 10 13 17

Pg. 484-487, # 1, 4, 6, 7, 10, 13, 17.

Independent Practice


9 3 quadratic inequalities in two variables

9.3 – quadratic inequalities in two variables

Chapter 9


Example8

example

Graph y < –2(x – 3)2 + 1

Determine if the point (2, –4) is a solution to the inequality.

  • Graph the parabola y = –2(x – 3)2 + 1

  • Draw it with a dotted line, since it’s “<“.

  • Choose two test points.

  • (0, 0) and (3, –3)

  • First point:

  • y < –2(x – 3)2 + 1

  • 0 < –2(0 – 3)2 + 1

  • 0 < –17

  • Doesn’t work.

  • Second point:

  • –3 < –2(3 – 3)2 + 1

  • –3 < 1

  •  Works.

The portion of the graph to be shaded is the portion that includes (3, –3).


Example9

example

Graph y < –2(x – 3)2 + 1

Determine if the point (2, –4) is a solution to the inequality.

  • b) We can see from the graph that the point is a solution to the inequality.

  • We should check algebraically:

  • y < –2(x – 3)2 + 1

  • –4 < –2(2 – 3)2 + 1

  • –4 < –2 + 1

  • –4 < –1

  • This is true, so the point is a solution to the inequality.


Try it1

Try it!

Graph y ≥ x2 – 4x – 5. Check if the point (–2, 4) is a solution to the inequality, both graphically and algebraically.


Example10

example

You can use a parabolic reflector to focus sound, light, or radio waves to a single point. A parabolic microphone has a parabolic reflector attached that directs incoming sounds to the microphone. René, a journalist, is using a parabolic microphone as he covers the Francophone Summer Festival of Vancouver. Describe the region that René can cover with his microphone if the reflector has a width of 50 cm and a max depth of 15 cm.


Example11

example

Samia and Jerrod want to learn the exhilarating sport of alpine rock climbing. They have enrolled in one of the summer camps at the Cascade Mountains in southern British Columbia. In the brochure, they come across an interesting fact about the manila rope that is used for rappelling down a cliff. It states that the rope can safely support a mass, M, in pounds, modelled by the inequality M ≤ 1450d2, where d is the diameter of the rope, in inches. Graph the inequality to examine how the mass that the rope supports is related to the diameter of the rope.


Pg 496 500 3 5 6 8 10 11 16

Pg. 496-500, #3, 5, 6, 8, 10, 11, 16.

Independent Practice


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