Browsing as Guest. Progress, bookmarks and attempts are disabled.
Log in to track your work.
0606 P22 - Jun 2022 - Q2 - 5 marks
7824
The diagram shows the graphs of \(y=|f(x)|\) and \(y=g(x)\), where \(y=f(x)\) and \(y=g(x)\) are straight lines. Solve the inequality \(|f(x)|\le g(x)\).
Solution
Answer: \(2\le x\le4\).
Answer: \(2\le x\le4\).
From the graph, the straight line \(f(x)\) has intercept 5 and gradient \(-2\), so
\(f(x)=-2x+5\).
The line \(g(x)\) has intercept \(-1\) and gradient 1, so
\(g(x)=x-1\).
The graph of \(|f(x)|\) is formed by reflecting the part of \(f(x)\) below the \(x\)-axis above the \(x\)-axis. Hence
\(|f(x)|=|-2x+5|\).
The critical points occur when
\(|-2x+5|=x-1\).
For \(x\le2.5\), \(|-2x+5|=-2x+5\). Then
\(-2x+5=x-1\),
so \(x=2\).
For \(x\ge2.5\), \(|-2x+5|=2x-5\). Then
\(2x-5=x-1\),
so \(x=4\).
Between these two intersection points, the graph of \(|f(x)|\) lies below or on the graph of \(g(x)\). Therefore
