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Functions, EXT1 EQ-Bank 18

A cubic function is given by  \(f(x)=\left(2 x^2+3 x-5\right)(x+2)\)

  1. Find the zeros of \(f(x)\).   (1 mark)

  2. Hence, or otherwise, solve \(f(x) \geqslant 0\), giving your answer in set notation.   (2 marks)

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a.    \(\text{Zeros at} \ \ x=-\dfrac{5}{2}, x=1 \ \ \text{and}\ \  x=-2\)

b.    \(x \in\left[-\frac{5}{2},-2\right] \cup\ x \in[1, \infty)\)

Show Worked Solution

a.    \(f(x)=\left(2 x^2+3 x-5\right)(x+2)=(2 x+5)(x-1)(x+2)\)

\(\text{Zeros at} \ \ x=-\dfrac{5}{2}, x=1 \ \ \text{and}\ \  x=-2\)
 

b.    \(\text{Find} \ x \ \text{such that} \ \ f(x) \geqslant 0.\)

\(\text{At} \ \ x=0: \ (5)(-1)(2)<0\)
 

\(\therefore \text{Graph}\ (f(x)) \geqslant 0 \ \ \text{for} \ \  x \in\left[-\frac{5}{2},-2\right] \cup\ x \in[1, \infty)\)

Functions, EXT1 EQ-Bank 12

Find all values of \(x\) for which  \((2 x+1)(x-3)(x-1) \leqslant 0\).   (2 marks)

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\(x \leqslant-\dfrac{1}{2}\ \cup \  1 \leqslant x \leqslant 3\)

Show Worked Solution

\((2 x+1)(x-3)(x-1) \leqslant 0\)

\(\text{Zeros at} \ \  x=-\dfrac{1}{2}, x=3 \ \ \text{and} \ \ x=1\)

\(\text{At} \ \ x=0: \ (1)(-3)(-1)>0\)
 

\(\therefore \text{Graph} \ \leqslant 0 \ \ \text{for} \ \ x \leqslant-\dfrac{1}{2}\ \cup \  1 \leqslant x \leqslant 3\)

Functions, EXT1 EQ-Bank 22

Solve the inequality  \(\left(2 x^2+3 x\right)(1-x) \geqslant 0\), expressing your answer in set notation.   (3 marks)

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\(x \in\left(-\infty,-\dfrac{3}{2}\right]\  \cup\  x \in [0,1] \)

Show Worked Solution

\(\left(2 x^2+3 x\right)(1-x)=x(2 x+3)(1-x)\)

\(x(2 x+3)(1-x) \geqslant 0\)

\(\text{Zeros at} \ \ x=0, x=-\dfrac{3}{2} \ \ \text{and} \ \ x=1\)

\(\text{At} \ \ x=-1: \ (-1)(1)(2)<0\)
 

\(\therefore \text{Graph} \geqslant 0 \ \ \text{for} \ \ x \in\left(-\infty,-\dfrac{3}{2}\right]\  \cup\  x \in [0,1] \)

Functions, EXT1 EQ-Bank 11

Solve the inequality  \((x-4)(x+2)(x-1) \geqslant 0\).   (2 marks)

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\(-2 \leqslant x \leqslant 1\  \cup\  x \geqslant 4\)

Show Worked Solution

\((x-4)(x+2)(x-1) \geqslant 0\)

\(\text{Zeros at} \ \ x=4, x=-2 \ \ \text{and}\ \  x=1\)

\(\text{At} \ \ x=0: \ (-4)(2)(-1)>0\)
 

\(\therefore \text{Graph}\ \geqslant 0 \ \text {for} \ -2 \leqslant x \leqslant 1\ \cup\ x \geqslant 4\)

Functions, EXT1 EQ-Bank 6 MC

For which values of \(x\) is  \((2x^2-3x-5)(x+2)<0\) ?

  1. \(x \in(-1,-2) \ \cup\  x \in\left(\dfrac{5}{2}, \infty\right]\)
  2. \(x \in[-\infty,-2)\  \cup\  x \in\left(-1, \dfrac{5}{2}\right)\)
  3. \(x \in(-1,-2)\  \cup\  x \in\left(\dfrac{5}{2}, \infty\right)\)
  4. \(x \in(-\infty,-2)\  \cup\  x \in\left(-1, \dfrac{5}{2}\right)\)
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\(D\)

Show Worked Solution

\((2x^2-3x-5)(x+2)=(2x-5)(x+1)(x+2) \)

\(\text {Zeros at} \ \ x=\dfrac{5}{2}, \ x=-1,\ \ \text {and} \ \ x=-2\)

\(\text{At} \ \ x=0: \ (-5)(1)(2)<0\)
 

\(\therefore \ \text{Graph }<0\ \text{ for }\ x \in(-\infty,-2)\  \cup\  x \in\left(-1, \frac{5}{2}\right) \)

\(\Rightarrow D\)

Functions, EXT1 EQ-Bank 2 MC

The inequality  \((2 x+4)(x-1)(3-x) \leqslant 0\)  has solution

  1. \(x \leqslant-2 \ \cup \  1 \lt x \leqslant 3\)
  2. \(-2 \leqslant x \leqslant 1 \ \cup \  x \geqslant 3\)
  3. \(x \leqslant-2 \ \cup \  x \geqslant 3\)
  4. \(-2 \leqslant x \leqslant 3\)
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\(B\)

Show Worked Solution

\(\text{Zeros at} \ \ x=-2, x=1 \ \ \text{and}\ \ x=3\)

\(\text{At} \ \ x=0: \ (4)(-1)(3)<0 \)
 

\(\therefore \ \text{Graph} \ \leqslant 0 \ \ \text{for} \ -2 \leqslant x \leqslant 1 \ \cup \ x \geqslant 3 \)

\(\Rightarrow B\)

Functions, EXT1 EQ-Bank 1 MC

What is the solution to the inequality

\((x-2)(x+1)(x-3)>0\)

  1. \(x<-1 \ \cup \ 2<x<3\)
  2. \(-1<x<2 \ \cup \ x>3\)
  3. \(x<-1 \ \cup \ x>3\)
  4. \(-1<x<3\)
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\(B\)

Show Worked Solution

\(\text {Zeros at}\ \  x=2, x=-1 \ \ \text {and} \ \ x=3\)

\(\text{At} \ \ x=0: \ (-2)(1)(-3)>0\)
 

\(\therefore \text {Graph}\ >0 \ \ \text{for}\ \ -1<x<2 \ \cup \ x>3\)

\(\Rightarrow B\)