ABCD is a rectangle with sides of lengths $x$ centimetres and $(x - 2)$ centimetres, as shown - Scottish Highers Maths - Question 8 - 2015
Question 8
ABCD is a rectangle with sides of lengths $x$ centimetres and $(x - 2)$ centimetres, as shown.
If the area of ABCD is less than $15 \, cm^2$, determine the range of... show full transcript
Worked Solution & Example Answer:ABCD is a rectangle with sides of lengths $x$ centimetres and $(x - 2)$ centimetres, as shown - Scottish Highers Maths - Question 8 - 2015
Step 1
Interpret Information
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Answer
To find the range of possible values for x, we first understand the relationship between the area of the rectangle and its dimensions. The area A of rectangle ABCD can be calculated as:
A=x(x−2)
According to the question, we want to find when the area is less than 15cm2:
x(x−2)<15
Step 2
Express in Standard Quadratic Form
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Answer
Rearranging the inequality gives us:
x2−2x−15<0
Step 3
Factorise
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Answer
Next, we factor the quadratic expression:
x2−2x−15=(x−5)(x+3)<0
Step 4
State Range
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Answer
To solve the inequality (x−5)(x+3)<0, we determine the critical points by setting the factors equal to zero:
x−5=0⇒x=5
x+3=0⇒x=−3
Now we can test intervals:
For x<−3, both factors are negative, so their product is positive.
For −3<x<5, the factor (x−5) is negative and (x+3) is positive, making the product negative.
For x>5, both factors are positive, and thus the product is positive.
Therefore, the solution to the inequality is:
−3<x<5
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