A trader at the local market loses money in any week when \(x^2 - 7x + 10\) takes a negative value. Here \(x\) is the number of hours the stall opens each d...
Question 1 Report
A trader at the local market loses money in any week when \(x^2 - 7x + 10\) takes a negative value. Here \(x\) is the number of hours the stall opens each day. Solve the inequality \(x^2 - 7x + 10 < 0\) to find the hours she should avoid. (3)
Answer Details
A quadratic inequality is solved in two stages: find where the expression equals zero, then decide which side of those values makes it negative.
- Factorise: \(x^2 - 7x + 10 = (x-2)(x-5)\), because \(-2\) and \(-5\) multiply to \(+10\) and add to \(-7\). The critical values are \(x = 2\) and \(x = 5\). [1]
- The coefficient of \(x^2\) is positive, so the graph is a parabola opening upwards. Such a curve dips below the horizontal axis only between its two roots. [1]
- Therefore \(x^2 - 7x + 10 < 0\) when \(2 < x < 5\). The trader loses money when the stall opens for between 2 and 5 hours a day, and should avoid that range. [1]
Test a value in each region to confirm: at \(x = 0\) the expression is \(+10\), at \(x = 3\) it is \(9 - 21 + 10 = -2\), and at \(x = 6\) it is \(36 - 42 + 10 = +4\). Only the middle region is negative.
The classic error is writing \(x < 2\) or \(x > 5\), which is the answer to the opposite inequality. Deciding the shape of the parabola first, rather than guessing, prevents that.
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