Quadratic equations

1. UNIVERSITY OF GUYANA
EMA 3102: BASIC
ALGEBRA AND GEOMETRY
LECTURER: MR. PETER WINTZ
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2. GROUP C MEMBERS
• Devishwar Bahadur
• Keith Damon
• Tanna James
• Judy Kendall
• Shanaz Hosain
• Ophela Johnson
• Shelana Cornelius
• Towana Johnson
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3. THE SOLUTION SET OF
LINEAR AND QUADRATIC
INEQUALITIES
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4. Presentation objectives
In this presentation, you will learn about:
• Linear inequalities
• Solving linear inequalities with one
variable
• Solving inequalities with two variables
• Quadratic Inequalities
• Solving Quadratic Inequalities
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5. Key Terms
• Variables:A symbol for a value we do not know as yet. It
is usually a letter like x or y.
• Inequality:Inequality refers to a relationship that makes
a non-equal comparison between two numbers or other
mathematical expressions.
• Equation:An equation is a mathematical expression that
contains an equal sign.
• Constant value: A constant is a fixed value.
• Linear: relating to, resembling, or having a graph that is
a line and especially a straight line
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6. Linear Inequalities
• Linear inequalities are defined as expressions in
which two linear expressions are compared
using the inequality symbols.
• Expressions in which two values are compared
using the inequality symbols.
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7. Symbols to represent linear
inequalities
Symbol Name Symbol Example
Not Equal ≠ x ≠ 5
Less than < x+9 < 15
Greater than > 3x +2 > 2x + 1
Less than or equal to ≤ x ≤ -6
Greater than or equal to ≥ y ≥ 2x+3
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8. Linear inequalities with one
variable
• The linear equations in one variable are
equations that are written as ax + b = 0, where
a, and b are two integers and x is a variable, and
there is only one solution.
• 8x + 3 = 8, for particular, is a linear equation with
only one variable. As a result, there is only one
solution to this equation.
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9. Example 1
Solve and graph the inequality: x > 2
Solution
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10. Graph
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11. Example 2
Solve and graph the inequality: x ≤ 3
Solution
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12. Graph
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13. Example 3
Solve and graph the inequality: x-5 >1
Solution
x-5>1
x-5>1
x > 5+1
x>6
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14. 14

15. Checking Answer
• x-5 >1
• x=8
8-5 >1
3 >1
TRUE
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16. Example 4
Solve and graph the inequality: 3x+1 ≤ 10
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17. Solution
3x+1 ≤ 10
3x ≤10-1
3x ≤9
x ≤ 9/3
x ≤3
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18. 18

19. Check Answer
• 3x+1 ≤ 10
• x=1
• 3(1)+1 ≤ 10
• 3+1 ≤ 10
• 4 ≤10
• TRUE
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20. Linear inequalities with two
variable
• A linear inequality in two variables is
formed when symbols other than equal to,
such as greater than or less than are used
to relate two expressions, and two
variables are involved.
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21. Example 1
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22. 22

23. Checking Answer
(5,-5)
• y>x-3
• -5>5-3
• -5>2
• FALSE
(2,5)
• y>x-3
• 5>2-3
• 5>2-3
• 5 >-1
• TRUE
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24. Example 2
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25. 25

26. Checking Answer
(-1,2)
• y≤2x-2
• 2≤2(-1)-2
• 2≤-2-2
• 2≤-4
• FALSE
(2,-4)
• y≤2x-2
• -4≤2(2)-2
• -4≤ 4-2
• -4 ≤ 2
• TRUE
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27. Quadratic Inequalities
• A quadratic inequality is an equation of second degree
that uses an inequality sign instead of an equal sign.
• The quadratic inequality has been derived from the
quadratic equation ax2 + bx + c = 0
• Further if the quadratic polynomial ax2 + bx + c is not
equal to zero, then they are either ax2 + bx + c > 0, or
ax2 + bx + c < 0, and are referred as quadratic
inequalities.
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28. Quadratic Inequalities
• Some examples of quadratic inequalities
in one variable are:
• x2 + x - 1 > 0
• 2x2 - 5x - 2 > 0
• x2 + 2x - 1 < 0
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29. Example 1
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Factors of10
X=10
+ = 7
5 2
Negative

30. Solution in interval notation: -5<x<-2
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x=2 x=-4 x=-6
(x+5) (x+2)< 0
(2+5) (2+2)<0
10x4<0
40<0
False
(x+5) (x+2)< 0
(-4+5) (-4+2)<0
1x-2<0
-2<0
True
(x+5) (x+2)< 0
(2+5) (2+2)<0
10+4<0
14<0
False

31. Example 2
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Factors of 16
X=16
+ = 10
8 2
Positive

32. `
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Solution in interval notation: -8>x>-2
x=-9 x=-5 x=1
(x+8) (x+2)< 0
(-9+8) (-9+2)<0
-1x-7<0
7>0
(x+8) (x+2)< 0
(-5+8) (-5+2)<0
3x-3<0
-9>0
(x+8) (x+2)< 0
(1+8) (1+2)<0
8x2<0
16>0

33. Example 3
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Factors of 8
X=8
+ = 9
8 1
Negative

34. • Solution in interval notation:-8≤x ≤-1
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35. THE END
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