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$x$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ |
$y$ | $-2$ | $0$ | $2$ | $4$ | $6$ | $8$ | $10$ |
The graph looks like this:
We need to find the area between $x=3$ and $x=7$, which is coloured green below.
The shape is a right-angled triangle, and its base is $4$ units and its height is $8$ units. So its area is $\frac{1}{2}\times4\times8=16$ square units.
Sketching the graph using relevant points
From the equation $y=2x-6$, we know that the graph will be a straight line.
Since we are interested in the area between $x=3$ and $x=7$, we could check where the graph is at those two points.
When $x=3$, $y=2\times3-6=6-6=0$.
When $x=7$, $y=2\times7-6=14-6=8$.
So $(3,0)$ and $(7,8)$ lie on the graph.
So the graph must be the straight line through those points, and the area required is the green area, as shown below.
The shape is a right-angled triangle, and its base is $4$ units and its height is $8$ units. So its area is $\frac{1}{2}\times4\times8=16$ square units.