Determine the value of x for which (x$^2$ - 1)>0

Analyze the question: We need all values of x for which x² − 1 is greater than zero.

Concept: To solve a quadratic inequality, factorise, find the critical points (where the expression equals zero), then test the regions on the number line.

Solution:

1. Factorise: x² − 1 = (x − 1)(x + 1)

2. Critical points: x = −1 and x = 1

3. These split the number line into three regions. Test one value in each:

| Region | Test value | (x − 1)(x + 1) | Sign | |---|---|---|---| | x < −1 | x = −2 | (−3)(−1) = 3 | positive ✔ | | −1 < x < 1 | x = 0 | (−1)(1) = −1 | negative ✘ | | x > 1 | x = 2 | (1)(3) = 3 | positive ✔ |

4. We want the expression to be greater than 0, so the solution is: x < −1 or x > 1

Checking the other options:

• B (−1 < x < 1) is where the expression is negative, so it is wrong.
• C (x > 0) includes x = 0.5, which gives 0.25 − 1 = −0.75, which is not positive. Wrong.
• D (x < −1) is only part of the answer. It misses x > 1.

Exam tip: For (x − a)(x − b) > 0 with a < b, the answer is "outside the roots" (x < a or x > b). For (x − a)(x − b) < 0, the answer is "between the roots" (a < x < b). A quick sketch of an upward-opening parabola will help you remember it: it is above the x-axis on the outer sides.