A positive square denominator still has a hole

\sqrt x\ge2x

Solution

(a)

Graphical inequalities

Use graphs to solve $\sqrt x\ge2x$ .

Explore this idea

Hint

Sketch $y=\sqrt x$ and $y=2x$ on the same axes.
Find where they meet.
The solution is where the square-root curve is on or above the line.

√x≥2x: two intersectionsScroll across to see the full diagram.

Answer · (a)

0\le x\le1/4

Explore the idea

Where is the square root above the line?

Switch on 'Solution' to shade the region between the curves from x = 0 to x = ¼.

Switch on a feature to add it to the sketch and see how it is found. A full-mark sketch shows all of them.

Y = √x
Y = 2x
Where the curve is above the line

Ideas to try

Check

x=0.1

\sqrt{0.1}=0.316\ge0.2

(b)

Rational inequalities

Solve $\dfrac{(x-1)(x-3)}{(x-2)^2}<0$ .

Explore this idea

Hint

The denominator $(x-2)^2$ is positive for every $x\ne2$ .
So the sign of the fraction is the sign of $(x-1)(x-3)$ .
But $x=2$ must be removed.

Answer · (b)

1<x<2

Explore the idea

A squared factor never changes the sign

The denominator (x − 2)² is positive on both sides of 2, so the chart is the same as for (x − 1)(x − 3), with a hole at 2.

\frac{(x-1)(x-3)}{(x-2)^2}<0

Choose an interval to test it. Shading marks the solution intervals.

Factor
x − 1	negative	positive	positive	positive
x − 3	negative	negative	negative	positive
(x − 2)²	positive	positive	positive	positive
product	positive	negative	negative	positive

x − 1	x − 3	(x − 2)²	product
negative	negative	positive	positive
positive	negative	positive	negative
positive	negative	positive	negative
positive	positive	positive	positive

Test value of xx = 2.50

Try an example

(x − 1)+(x − 3)−((x − 2)²)+⇒ product negative

Is it < 0?

Yes

x is in the solution set

The shaded strips are where the expression satisfies the inequality; they match the green-tinted columns of the chart.

Expression, and the x-values where it is negative
Your test value

Key idea

A repeated (even-power) factor never changes the sign, but it still creates a point where the expression does not exist.

Ideas to try

Drag the test value across $x=2$ . Does the sign change?
Test exactly $x=2$ .

A positive square denominator still has a hole

Solution

Graphical inequalities

Where they meet

Compare

Where is the square root above the line?

Rational inequalities

Write down what $x$ cannot be

Use the positive denominator

Solve the quadratic inequality

Remove the excluded point