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A-Level Equations and inequalities
What the A-Level syllabus expects for Equations and inequalities, and how to practise it.
What the syllabus expects
- Building an equation, a linear system, or inequalities out of a described situation
- Finding exact or approximate solutions to an equation with a graphing calculator or graphing software
- Using a graphing calculator or graphing software to solve a system of linear equations
- Solving rational inequalities f(x)/g(x) > 0 in which each of f(x) and g(x) is a linear or quadratic expression
- Tackling inequalities through graphical approaches
How it's examined
Questions on this topic most often ask you to solve, sketch, find, describe. About 3% of the past-paper style questions in Rae's bank for this subject sit in this topic.
Worked examples
Example 1 (4 marks)
Without a calculator, solve (x^2 - 5x + 6)/(x^2 - 4) < (2x - 3)/(x + 2).
Show the worked answer
Factor: (x²-5x+6)/(x²-4) = (x-2)(x-3)/[(x-2)(x+2)] = (x-3)/(x+2) for x != 2. Inequality becomes (x-3)/(x+2) < (2x-3)/(x+2). Bring together: [(x-3)-(2x-3)]/(x+2) < 0, i.e. (-x)/(x+2) < 0, i.e. x/(x+2) > 0, giving x < -2 or x > 0. Exclude x = 2 (undefined in original). Note x = -2 already excluded.
Example 2 (3 marks)
The curve C is given by y = ln(x - 1) + 2. By drawing a suitable straight line together with the graph of C, find the solution of the equation ln(x - 1) + 7 = 2x.
Show the worked answer
C: y = ln(x-1) + 2. The equation ln(x-1) + 7 = 2x rearranges to ln(x-1) + 2 = 2x - 5, so draw the straight line y = 2x - 5. Its intersections with C give the solutions. From the graph (GC), the x-coordinates of intersection are x = 1.01 and x = 4.06 (3sf).
Example 3 (2 marks)
Use the previous sketch to solve the inequality ln|x| < |x| - 5.
Show the worked answer
Both sides are even in x, so solve for x>0: ln x < x - 5. The critical values are the roots of ln x = x - 5. Using the graph (GC), these are x = 0.00678 and x = 6.94 (3sf). Testing x=1 gives ln1=0 and 1-5=-4, so 0 > -4, i.e. ln x > x-5 between the roots; hence ln x < x-5 for x < 0.00678 or x > 6.94 (x>0). By symmetry (replace x by |x|): 0 < |x| < 0.00678 or |x| > 6.94.
More worked questions on this topic
- Using part (a) or otherwise, solve |x - a| > 1/|x - a| for a real constant a > 1. (3 marks)
- Curve C is y=(1+3x−18x²)/(9x+3). Without a calculator, find the set of values y can take. (4 marks)
- On one set of axes, sketch y = (x+1)/(2x+3) and y = x^2 + x − 3, marking clearly the equation(s (5 marks)
- Pure Mathematics. Green Harvest Market was running a promotion on several grocery items. Organi (4 marks)
- (a) Find the set of x with 3|x - 3| <= |1 - 2x|. (b) Without a calculator, solve (x + 18)/(x^2 (6 marks)
- (i) In terms of a, solve 1/(x-a)²=|x-a|. (ii) On one set of axes, sketch y=1/(x-a)² and y=|x-a| (6 marks)
More A-Level H2 Maths topics
Functions · Graphs and their transformations · Sequences and series · Vectors in two and three dimensions: basic properties · Scalar and vector products · Vector geometry in three dimensions · all of A-Level H2 Maths