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2014 Paper 2 Q7
D: 1600.0
B: 1486.9
modulus function
curve sketching
graphical solution
piecewise linear
number of solutions
inequality
quadrilateral
absolute value equations
- The function \(\f\) is defined by \(\f(x)= |x-a| + |x-b| \), where \(a < b\). Sketch the graph of \(\f(x)\), giving the gradient in each of the regions \(x < a\), \(a < x < b\) and \(x > b\). Sketch on the same diagram the graph of \(\g(x)\), where \(\g(x)= |2x-a-b|\). What shape is the quadrilateral with vertices \((a,0)\), \((b,0)\), \((b,\f(b))\) and \((a, \f(a))\)?
- Show graphically that the equation \[ |x-a| + |x-b| = |x-c|\,, \] where \(a < b\), has \(0\), \(1\) or \(2\) solutions, stating the relationship of \(c\) to \(a\) and \(b\) in each case.
- For the equation \[ |x-a| + |x-b| = |x-c|+|x-d|\,, \] where \(a < b\), \(c < d\) and \(d-c < b-a\), determine the number of solutions in the various cases that arise, stating the relationship between \(a\), \(b\), \(c\) and \(d\) in each case.
Solution:
- \(\,\)
\((a,0)\), \((b,0)\), \((b,\f(b))\) and \((a, \f(a))\) forms a rectangle.
- There are no solutions if \(a < c < b\):
There is one solution if \(a=c\) or \(a = b\)
And there are two solution if \(c \not \in [a,b]\)
There is exactly one solution unless....
... there are infinitely many solutions when the gradients line up perfectly, ie when \(a+b=c+d\)
