The variables x and y are connected by the equation $y = 2x + \dfrac{6}{x} - 7$. Some corresponding values of x and y are given in the table below.

1. Identify the x-value for which the y-value is a and substitute it into the equation.

From the table, $a$ is the value of $y$ when $x=2$.

$a=2(2)+\frac{6}{2}-7$
$=4+3-7$
$=0$

1. Use the completed table of values for plotting.

From part (a), $a=0$, so the point for $x=2$ is $(2,0)$.

The points to plot are:
$(0.5,6),\ (1,1),\ (1.5,0),\ (2,0),\ (3,1),\ (4,2.5),\ (5,4.2),\ (6,6),\ (7,7.9).$

2. Plot the points accurately and draw the curve over the required interval.

Plot all the points on the given grid.

Join the points with a smooth curve from $x=0.5$ to $x=7$, without using straight line segments between successive points.

1. Draw the tangent through the given external point so that it just touches the curve.

On the completed graph, mark $A(3,-1)$. Place a ruler through $A$ and adjust it until the straight line just touches the curve at one point. Draw this line.

The point of contact is approximately $(1.5,0)$, so the tangent should pass through $A(3,-1)$ and touch the curve near $(1.5,0)$.

1. Find the gradient of the tangent using two points on the tangent.

Using the points $(1.5,0)$ and $A(3,-1)$,
$m=\frac{-1-0}{3-1.5}=\frac{-1}{1.5}=-\frac{2}{3}.$

2. Use the gradient and a point on the tangent to find the equation of the line.

Use $y=mx+c$ and substitute $m=-\dfrac{2}{3}$ and $(3,-1)$:
$-1=-\frac{2}{3}(3)+c$
$-1=-2+c$
$c=1.$
Hence the tangent is
$y=-\frac{2}{3}x+1.$

1. Rewrite the given equation in a form involving the plotted curve.

Since $0.5\le x\le 7$, $x\ne 0$, so we may divide by $x$.

$3x^2-8x+6=0$
$3x-8+\frac{6}{x}=0$
$3x+\frac{6}{x}-8=0$

2. Connect the equation to the curve and identify the straight line to draw.

The curve already drawn is

$y=2x+\frac{6}{x}-7.$

For the quadratic equation,

$3x+\frac{6}{x}-8=0$

can be written as

$2x+\frac{6}{x}-7=1-x.$

So the suitable straight line to draw is

$y=1-x.$

3. Use the graph to justify that there is no solution in the interval.

Draw the straight line $y=1-x$ on the same grid.

From the graph, the line $y=1-x$ does not meet the curve $y=2x+\dfrac{6}{x}-7$ for $0.5\le x\le 7$.

Therefore there is no value of $x$ in this interval satisfying

$3x^2-8x+6=0.$