Theory question 107
Inequalities(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.
Show the model answer
Model answer
(a)(i) 4x < 2 + 3x gives x < 2. x − 8 < 3x gives −8 < 2x, so x > −4. Together: −4 < x < 2.
(ii) Number line with open circles at −4 and 2 and the segment between them marked.
(b) Let the bill be 10t + u with t + u = 9. (10u + t) − (10t + u) = 45 gives 9(u − t) = 45, so u − t = 5.
Solving: u = 7, t = 2. The bill should be £27.
Explanation
Use open circles because the inequalities are strict.