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Sketch the graph of y = 4 - |2x - 6| Solve the inequality Solve the inequality 4 - |2x - 6| > 2 - AQA - A-Level Maths Pure - Question 4 - 2020 - Paper 1
Question 4
Sketch the graph of y = 4 - |2x - 6| Solve the inequality Solve the inequality 4 - |2x - 6| > 2
Worked Solution & Example Answer:Sketch the graph of y = 4 - |2x - 6| Solve the inequality Solve the inequality 4 - |2x - 6| > 2 - AQA - A-Level Maths Pure - Question 4 - 2020 - Paper 1
Step 1
Sketch the graph of y = 4 - |2x - 6|
Answer
To sketch the graph of the function, we first need to understand its structure. The expression inside the absolute value is ( |2x - 6| ), which shifts the graph left or right.
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Find the Vertex: Start by setting the inside of the absolute value to zero: [ 2x - 6 = 0 ] [ 2x = 6 ] [ x = 3 ] Plugging ( x = 3 ) back into the equation: ( y = 4 - |2(3) - 6| = 4 ). Thus, the vertex is at ( (3, 4) ).
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Identify Intercepts: The graph will intersect the y-axis when ( x = 0 ):
[ y = 4 - |2(0) - 6| = 4 - 6 = -2 ]
So one point is ( (0, -2) ). To find the x-intercepts, set ( y = 0 ):
[ 0 = 4 - |2x - 6| ]
[ |2x - 6| = 4 ]
This gives two cases:- Case 1: ( 2x - 6 = 4 \implies 2x = 10 \implies x = 5 )
- Case 2: ( 2x - 6 = -4 \implies 2x = 2 \implies x = 1 )
Thus, the x-intercepts are at ( (1, 0) ) and ( (5, 0) ).
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Sketch the Graph: Plot the points ( (3, 4) ), ( (0, -2) ), ( (1, 0) ), and ( (5, 0) ) on a coordinate plane. The graph is an inverted V shape, opening downwards with vertex at ( (3, 4) ).
Step 2
Solve the inequality 4 - |2x - 6| > 2
Answer
To solve the inequality, start by isolating the absolute value:
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Rearranging the inequality: [ 4 - |2x - 6| > 2 ] [ -|2x - 6| > -2 ] (Multiply by -1, reversing the inequality) [ |2x - 6| < 2 ]
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Removing the absolute value: This results in two inequalities to solve:
- Case 1: ( 2x - 6 < 2 )
[ 2x < 8 \implies x < 4 ] - Case 2: ( 2x - 6 > -2 )
[ 2x > 4 \implies x > 2 ]
- Case 1: ( 2x - 6 < 2 )
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Combine the solutions: Thus, the solution to the inequality is: [ 2 < x < 4 ]
This is the final answer.
Step 3
Solve the inequality 4 - |2x - 6| > 2
Answer
To solve ( 4 - |2x - 6| > 2 ), start by isolating the absolute value:
[ -|2x - 6| > -2 ] This simplifies to: [ |2x - 6| < 2 ]
Next, split it into two inequalities:
- ( 2x - 6 < 2 \rightarrow 2x < 8 \rightarrow x < 4 )
- ( 2x - 6 > -2 \rightarrow 2x > 4 \rightarrow x > 2 )
Combining gives: [ 2 < x < 4 ]