Which diagram best represents the solution set of $x^2 - 2x - 3 \geq 0$?
A - HSC - SSCE Mathematics Extension 1 - Question 1 - 2020 - Paper 1
Question 1
Which diagram best represents the solution set of $x^2 - 2x - 3 \geq 0$?
A.
B.
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Worked Solution & Example Answer:Which diagram best represents the solution set of $x^2 - 2x - 3 \geq 0$?
A - HSC - SSCE Mathematics Extension 1 - Question 1 - 2020 - Paper 1
Step 1
Solve the inequality $x^2 - 2x - 3 \geq 0$
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Answer
First, we can rewrite the equation as follows:
x2−2x−3=0
Next, we factor the quadratic expression:
(x−3)(x+1)=0
Setting each factor equal to zero gives us the critical points:
x−3=0⇒x=3
x+1=0⇒x=−1
These critical points will help us determine the intervals to test for the inequality.
Step 2
Test intervals
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Answer
The critical points split the number line into three intervals:
(−∞,−1)
(−1,3)
(3,∞)
We can test one point from each interval to determine where the inequality holds:
For the interval (−∞,−1), choose x=−2:
(−2)2−2(−2)−3=4+4−3=5≥0 (True)
For the interval (−1,3), choose x=0:
02−2(0)−3=−3<0 (False)
For the interval (3,∞), choose x=4:
42−2(4)−3=16−8−3=5≥0 (True)
From this testing, we find that the solution set is intervals (−∞,−1] and [3,∞).
Step 3
Select the correct diagram
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Answer
The diagrams representing the solution set (−∞,−1]∪[3,∞) must include shaded regions at −1 and 3 with arrows continuing to the left and right. Therefore, the correct choice is:
