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Solve the inequality - OCR - GCSE Maths - Question 17 - 2017 - Paper 1

Question 17

Solve the inequality. $$x^2 - 5x - 6 < 0$$

Worked Solution & Example Answer:Solve the inequality - OCR - GCSE Maths - Question 17 - 2017 - Paper 1

Step 1

Factor the quadratic expression

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Answer

To solve the inequality, we first factor the quadratic expression. We need to find two numbers that multiply to -6 and add up to -5. The numbers -6 and 1 meet these criteria.

Thus, the equation can be factored as: $(x - 6)(x + 1) < 0$

Step 2

Determine the critical points

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Answer

Setting the factors equal to zero gives the critical points:

$x - 6 = 0 \Rightarrow x = 6$
$x + 1 = 0 \Rightarrow x = -1$

These points will help to identify the intervals to test.

Step 3

Test intervals

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Answer

Next, we test the intervals determined by our critical points:

For the interval $(-\infty, -1)$ , choose $x = -2$ : $(-2 - 6)(-2 + 1) = (-8)(-1) = 8 > 0$
For the interval $(-1, 6)$ , choose $x = 0$ : $(0 - 6)(0 + 1) = (-6)(1) = -6 < 0$
For the interval $(6, \infty)$ , choose $x = 7$ : $(7 - 6)(7 + 1) = (1)(8) = 8 > 0$

Step 4

Write the solution

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Answer

The inequality is satisfied for the interval where the product is less than zero, which is:

$-1 < x < 6$

Thus, the final solution to the inequality is: $x \in (-1, 6)$

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