Quadratic Inequalities

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Method ofGraph sketching
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Example 1
Solve the quadratic inequality x2 5x 6 gt 0
graphically.
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Procedures
The corresponding quadratic function is y
x2 5x 6
Step (1)
Step (2)
Factorize x2 5x 6,
we have y (x 2)(x 3) ,i.e. y 0, when x
2 or x 3.
Sketch the graph of y x2 5x 6.
Step (3)
Step (4)
Find the solution from the graph.
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Sketch the graph y x2 5x 6 .
What is the solution of x2 5x 6 gt 0 ?
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We need to solve x 2 5x 6 gt 0,
y
The portion of the graph above the x-axis
represents y gt 0 (i.e. x 2 5x 6 gt 0)
x
0
The portion of the graph below the x-axis
represents y lt 0 (i.e. x 2 5x 6 lt 0)
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y
When x lt 2, the curve is above the x-axis i.e., y
gt 0 x2 5x 6 gt 0
When x gt 3, the curve is above the x-axis i.e., y
gt 0 x2 5x 6 gt 0
x
0
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From the sketch, we obtain the solution

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Graphical Solution
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Example 2
Solve the quadratic inequality x2 5x 6 lt 0
graphically.
Same method as example 1 !!!
10
y
x
0
When 2 lt x lt 3, the curve is below the
x-axis i.e., y lt 0 x2 5x 6 lt 0
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From the sketch, we obtain the solution

2 lt x lt 3
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Graphical Solution
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Exercise 1
Find the x-intercepts of the curve (x 2)(x
1)0 x 2 or x 1

• Solve

Answer
x lt 2 or x gt 1
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Exercise 2
Find the x-intercepts of the curve x2 x 12
0 (x 3)(x 4)0 x 3 or x 4

• Solve

Answer
3 lt x lt 4
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Find the x-intercepts of the curve (x 7)(x
5)0 x 7 or x 5
Exercise 3
Solve
Solution
7 lt x lt 5
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Find the x-intercepts of the curve (x 3)(3x
2)0 x 3 or x 2/3
Exercise 4
Solve
Solution
x ? 3 or x ? 2/3