A repeated factor does not disappear from a strict inequality

x^2(x+2)^2-1=(x+1)^2(x+1+\sqrt2)(x+1-\sqrt2)

Total marks: 6

Solution

(i)

Factorising

Show that $x^2(x+2)^2-1=(x+1)^2(x+1+\sqrt2)(x+1-\sqrt2)$ .

Hint

Write $x(x+2)=x^2+2x$ so the left side is a square minus 1.
Difference of two squares twice.

Answer · (i)

x^2(x+2)^2-1=(x+1)^2(x+1+\sqrt2)(x+1-\sqrt2)

(ii)

Rational inequalities

Hence solve $\dfrac1{(x+2)^2}>x^2$ without a calculator.

Explore this idea

Hint

State what $x$ cannot be: $x\ne-2$ .
$(x+2)^2>0$ , so multiply by it safely.
Reduce to the factorised quartic from (i).

Answer · (ii)

-1-\sqrt2<x<-1+\sqrt2

Explore the idea

A double factor and a double pole

Both (x + 1)² and (x + 2)² keep their sign, but the points x = −1 and x = −2 are still special.

\frac{(x+1)^2(x+1+\sqrt2)(x+1-\sqrt2)}{(x+2)^2}<0

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

Factor
(x + 1)²	positive	positive	positive	positive	positive
x + 1 + √2	negative	positive	positive	positive	positive
x + 1 − √2	negative	negative	negative	negative	positive
(x + 2)²	positive	positive	positive	positive	positive
product	positive	negative	negative	negative	positive

(x + 1)²	x + 1 + √2	x + 1 − √2	(x + 2)²	product
positive	negative	negative	positive	positive
positive	positive	negative	positive	negative
positive	positive	negative	positive	negative
positive	positive	negative	positive	negative
positive	positive	positive	positive	positive

Test value of xx = -1.50

Try an example

((x + 1)²)+(x + 1 + √2)+(x + 1 − √2)−((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 where it is negative
Your test value

Key idea

Even-power factors keep the sign on both sides, but they carve a hole at their root. Three intervals result, not one.

Ideas to try

Test $x=-1$ and $x=-2$ . What goes wrong at each?
How many separate intervals does the solution have?

(iii)

Exponential substitution

Deduce the values of $x$ satisfying $\dfrac1{(e^x+2)^2}>e^{2x}$ .

Explore this idea

Hint

This is (ii) with $x$ replaced by $e^x$ .
$e^x>0$ , so only the positive part of the solution survives.

Answer · (iii)

x<\ln(\sqrt2-1)

Explore the idea

e^x below √2 − 1

Between y = 0 and y = √2 − 1 ≈ 0.414 the exponential curve lies to the left of x = ln(√2 − 1) ≈ −0.88.

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 = eˣ
Y = √2 − 1
Boundary

Ideas to try

Why does the lower bound

-1-\sqrt2

not matter?

A repeated factor does not disappear from a strict inequality

Solution

Factorising

A difference of squares

Recognise a square

Another difference of squares

Rational inequalities

Exclude the zero of the denominator

Clear the positive denominator

Use part (i)

The repeated factor

Solve and exclude

A double factor and a double pole

Exponential substitution

Substitute

Keep only positive values

Take logarithms

e^x below √2 − 1