(a)(i) If \(4x < 2 + 3x\) and \(x – 8 < 3x\), what range of values of x satisfies both inequalities? ; (ii) Represent your result in (i) on the number line.
(b) A shop is sending out a bill for an amount less than £100. The accountant interchanges the two digits and so overcharges the customer by 45. Given that the sum of the two digits is 9, find how much the bill should be.
Explanation
(a)(i) \(4x < 2 + 3x ..... (1)\)
\(4x - 3x < 2\)
\(x < 2\)
\(x - 8 < 3x ..... (2)\)
\(-8 < 2x\)
\(x > -4\)
\(x < 2 ; x > -4\).
(ii) 
(b) Let the digits of the number be x and y.
\(x + y = 9 .... (1)\)
\((10y + x) - (10x + y) = 45 \)
\(10y - y + x - 10x = 45\)
\(9y - 9x = 45 .... (2)\)
From (1), y = 9 - x. Putting it into (2), we have
\(9(9 - x) - 9x = 45\)
\(81 - 9x - 9x = 45\)
\(36 = 18x \implies x = 2\)
\(\therefore y = 9 - x\)
= \(9 - 2 = 7\)
\(\therefore\) The bill = £27.