Find all x such that \(x^2+5x<6\). Express your answer in interval notation.
+9466 x2 + 5x < 6
Subtract 6 from both sides of the inequality.
x2 + 5x - 6 < 0
Let's find what values of x make x2 + 5x - 6 equal 0
x2 + 5x - 6 = 0
Factor the left side. What two numbers add to 5 and multiply to -6 ? -1 and +6
(x - 1)(x + 6) = 0
Set each factor equal to zero and solve for x
| x - 1 = 0 or x + 6 = 0 |
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| x = 1 x = -6 |
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| Since a graph of y = x2 + 5x - 6 is a parabola, we can be sure that |
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| the values of x that would make y < 0 fall in one of these two intervals: |
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| either the interval (-6, 1) or the interval (-∞, -6) U (1, ∞) |
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| To determine which interval is the solution set, let's test a number in both of them. |
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| 0 is a number in the interval (-6, 1) |
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| If x = 0 then y = (0)2 + 5(0) - 6 = -6 |
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| And -6 < 0 so we know 0 should be included. |
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| 2 is a number in the interval (-∞, -6) U (1, ∞) |
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| If x = 2 then y = (2)2 + 5(2) - 6 = 8 |
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| And 2 > 0 so we know 2 should not be included. |
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| So we can be sure that x2 + 5x - 6 < 0 if and only if x is in the interval (-6, 1) |
