Solve absolute value inequality

1 Solve the inequality \(| 3 x - 1 | < 2\).

1 Solve the inequality \(| 9 - 2 x | < 1\).

2 Solve the inequality \(| 2 x - 3 | \leqslant | 3 x |\).

1 Solve the inequality \(| 3 x - 5 | < 2 | x |\).

1. A physics student is studying the movement of particles in an electric field. In one experiment, the distances in micrometres of two moving particles, \(A\) and \(B\), from a fixed point \(O\) are modelled by

$$\begin{aligned} & d _ { A } = | 5 t - 31 | \\ & d _ { B } = \left| 3 t ^ { 2 } - 25 t + 8 \right| \end{aligned}$$ respectively, where \(t\) is the time in seconds after motion begins.

1. Use algebra to find the range of time for which particle \(A\) is further away from \(O\) than particle \(B\) is from \(O\). It was recorded that the distance of particle \(B\) from \(O\) was less than the distance of particle \(A\) from \(O\) for approximately 4 seconds.
2. Use this information to assess the validity of the model.

3. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = \frac { x } { | x | - 2 }$$ Use algebra to determine the values of \(x\) for which $$2 x - 5 > \frac { x } { | x | - 2 }$$

7. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with equation \(y = \left| x ^ { 2 } - 8 \right|\) and a sketch of the straight line with equation \(y = m x + c\), where \(m\) and \(c\) are positive constants. The equation $$\left| x ^ { 2 } - 8 \right| = m x + c$$ has exactly 3 roots, as shown in Figure 1.

1. Show that $$m ^ { 2 } - 4 c + 32 = 0$$ Given that \(c = 3 m\)
2. determine the value of \(m\) and the value of \(c\)
3. Hence solve $$\left| x ^ { 2 } - 8 \right| \geqslant m x + c$$

1. Use algebra to determine the values of \(x\) for which

$$\left| x ^ { 2 } - 2 x \right| \leqslant x$$

3 The set \(\mathcal { A }\) is defined by \(\mathcal { A } = \{ x : - \sqrt { } 2 < x < 0 \} \cup \{ x : 0 < x < \sqrt { } 2 \}\) Which of the inequalities given below has \(\mathcal { A }\) as its solution?
Circle your answer.
[0pt] [1 mark] \(\left| x ^ { 2 } - 1 \right| > 1\) \(\left| x ^ { 2 } - 1 \right| \geq 1\) \(\left| x ^ { 2 } - 1 \right| < 1\) \(\left| x ^ { 2 } - 1 \right| \leq 1\)