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

1 Solve \(5 x + 3 < | 3 x - 1 |\).

Solve the inequality \(|5x + 7| > |2x - 3|\). [4]

Solve the inequality \(|x - 7| > 4x + 3\). [4]

Find the set of values of \(x\) satisfying the inequality \(|8 - 3x| < 2\). [3]

Solve the inequality \(|x - 3| < 3x - 4\). [4]

Find the set of values of \(x\) for which $$|x^2 - 4| > 3x.$$ [5]

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

Solve the inequality \(|4x - 3| < |2x + 1|\). [5]

Find the set of values of \(x\) such that $$|3x + 1| \leq |x - 2|.$$ [5]

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

The diagram shows the line \(y = 5 - x\) \includegraphics{figure_15}

1. On the diagram above, sketch the graph of \(y = |x^2 - 4x|\), including all parts of the graph where it intersects the line \(y = 5 - x\) (You do not need to show the coordinates of the points of intersection.) [3 marks]
2. Find the solution of the inequality $$|x^2 - 4x| > 5 - x$$ Give your answer in an exact form. [4 marks]

Solve the following inequalities giving your answer in set notation:

1. \(|4x + 3| < |x - 8|\) [3]
2. \(\frac{x}{x^2+1} < \frac{1}{2}\) [3]