(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....
Question 1 Report
(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.
Answer Details
(a)(i) Solve both inequalities.
\(4x < 2 + 3x \ \Rightarrow\ x < 2\)
\(x - 8 < 3x \ \Rightarrow\ -8 < 2x \ \Rightarrow\ x > -4\)
Both hold when \(\mathbf{-4 < x < 2}\).
(ii) Number line: an open circle at \(-4\) and an open circle at \(2\), with the segment between them shaded (values strictly between, endpoints excluded).
(b) Let the correct bill be the two-digit number \(10a + b\) (\(a\) = tens digit, \(b\) = units digit). Interchanging gives \(10b + a\), which overcharges by 45:
\((10b + a) - (10a + b) = 45 \ \Rightarrow\ 9(b - a) = 45 \ \Rightarrow\ b - a = 5\)
Also \(a + b = 9\). Adding: \(2b = 14 \ \Rightarrow\ b = 7\), so \(a = 2\).
The bill should be \(10a + b = \mathbf{£27}\). (Check: \(72 - 27 = 45\).)
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