Grade 10 Maths · Functions

Solve product inequality from graphs

Use two graphs to find where f(x) × g(x) or f(x) ÷ g(x) is positive or negative by comparing their signs on each interval.

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for which values of x is f(x).g(x)≥0. functions grade 12Benit mathematics

For which values of x is f(x).g(x) greater than zero | GRAPH INTERPRETATION |GRADE 12,11,10PushProgramme

For which values of x will x times f at x be greater than zero. Functions grade 12, 11 and 10Benit mathematics

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Question 9.3 · May/June 2024 · Paper 1 · National (DBE) · 3 marks

The graph of

f (x) = a x^{3} + b x^{2} + c x - 5

is drawn below.

E (- 1; 0)

and

G (5; 0)

are the

x

-intercepts of

f

9.3 Use the graph to determine the values of

x

for which

f^{''} (x) \cdot f (x) > 0

.(3)

Work it out on paper first. Then mark yourself with the official memo, one tick at a time.

More past paper questions: solve product inequality from graphs

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3. Question 7.4.1 · May/June 2025 · Paper 2 · National (DBE) · 2 marks
In the diagram, the graphs of $f (x) = 2 cos x + 1$ and $g (x) = sin 2 x$ are drawn for the interval $x \in [- 18 0^{\circ}; 18 0^{\circ}]$ .
7.4 Use the graphs to determine the values of x, in the interval $x \in [- 18 0^{\circ}; 18 0^{\circ}]$ for which:
7.4.1 $g (x) \cdot f^{'} (x) < 0$ (2)
Show the memo (2 ticks)Memo
1. ✓
  Analyze the conditions for the product to be negativeFor $g (x) \cdot f^{'} (x) < 0$ , the signs of $g (x)$ and $f^{'} (x)$ must be different (+ and - OR - and +).
  $f^{'} (x)$ represents the gradient of $f$ . It is positive when $f$ is increasing ( $- 18 0^{\circ} < x < 0^{\circ}$ ) and negative when $f$ is decreasing ( $0^{\circ} < x < 18 0^{\circ}$ ).
  
  Condition 1: $g (x) > 0$ and $f^{'} (x) < 0$
  $g (x)$ is positive for $(0^{\circ}; 9 0^{\circ}) \cup (- 18 0^{\circ}; - 9 0^{\circ})$ .
  $f^{'} (x)$ is negative for $(0^{\circ}; 18 0^{\circ})$ .
  Intersection: $x \in (0^{\circ}; 9 0^{\circ})$
  
  Condition 2: $g (x) < 0$ and $f^{'} (x) > 0$
  $g (x)$ is negative for $(9 0^{\circ}; 18 0^{\circ}) \cup (- 9 0^{\circ}; 0^{\circ})$ .
  $f^{'} (x)$ is positive for $(- 18 0^{\circ}; 0^{\circ})$ .
  Intersection: $x \in (- 9 0^{\circ}; 0^{\circ})$
  1 mark · Solve product inequality from graphs
2. ✓
  State final union of intervals $x \in (- 9 0^{\circ}; 0^{\circ}) \cup (0^{\circ}; 9 0^{\circ})$
  
  OR
  
  $- 9 0^{\circ} < x < 0^{\circ}$ or $0^{\circ} < x < 9 0^{\circ}$
  1 mark · Solve product inequality from graphs
4. Question 6.2.3 · March 2025 · Provincial · 2 marks
6.2 In the diagram, the graphs of $f (x) = sin 2 x$ and $g (x) = cos (x - 4 5^{\circ})$ are drawn for the interval $x \in [- 18 0^{\circ}; 18 0^{\circ}]$ .
6.2.3 Use the graphs to determine the values of $x$ in the interval $x \in [0^{\circ}; 18 0^{\circ}]$ for which $\frac{f ( x )}{g ( x )} < 0$ .(2)
Show the memo (1 tick)Memo
1. ✓
  Identify interval where signs differ $9 0^{\circ} < x < 13 5^{\circ}$
  2 marks · Solve product inequality from graphs
5. Question 5.5 · November 2025 · Paper 1 · Provincial · 2 marks
The graphs of $f (x) = \frac{a}{x + p} + q$ and $g (x) = \frac{x}{3} + c$ are drawn below. C $(3; - 1)$ is the point of intersection between the asymptotes of $f$ . A and B are $x$ - and $y$ -intercepts of $f$ and $g$ respectively. E $(4; 2)$ is a point on $f$ in the first quadrant. $Diagram for this question$
5.5 Determine the value of $x$ , for which $f (x) \cdot g (x) < 0$ (2)
Show the memo (1 tick)Memo
1. ✓
  Answer $x > 3$ , $x \neq = 6$
  OR $3 < x < 6$ or $x > 6$
  2 marks · Solve product inequality from graphs
6. Question 3.4.2 · June 2024 · Paper 2 · Provincial · 2 marks
In the diagram below, the graphs of $f (x) = a cos x$ and $g (x) = sin (x + p)$ are drawn for $x \in [- 18 0^{\circ}; 18 0^{\circ}]$ .
3.4 Determine the value(s) of $x$ if:
3.4.2 $x \cdot f (x) > 0$ (2)
Show the memo (2 ticks)Memo
1. ✓
  Negative x-values $x < 0$ and $f (x) < 0$ : $- 9 0^{\circ} < x < 0^{\circ}$
  1 mark · Solve product inequality from graphs
2. ✓
  Positive x-values $x > 0$ and $f (x) > 0$ : $9 0^{\circ} < x \leq 18 0^{\circ}$
  1 mark · Solve product inequality from graphs

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