H2 Maths 1.3 - Rational Inequality via Sign Chart

Step 1 - Set up the problem

We want to solve the inequality three minus x over x squared minus four, strictly less than zero.

Step 1 - Set up the problem

The key idea: a fraction is negative when its numerator and denominator have opposite signs.

Step 1 - Set up the problem

So we need to track the sign of each factor separately.

Step 2 - Find the critical points

Critical points are where the expression equals zero or is undefined.

Step 2 - Find the critical points

The numerator three minus x equals zero when x equals three.

Step 2 - Find the critical points

Factor the denominator: x squared minus four is x minus two times x plus two.

Step 2 - Find the critical points

So the denominator is zero at x equals two and x equals negative two.

Step 2 - Find the critical points

These two are excluded from the solution - the expression is undefined there.

Step 3 - Build the sign chart

The three critical points divide the number line into four intervals.

Step 3 - Build the sign chart

We test the sign of each factor - three minus x, x minus two, and x plus two - in each interval.

Step 3 - Build the sign chart

Then multiply the signs to find the sign of the whole fraction.

Step 4 - Read off where the fraction is negative

We want the fraction strictly less than zero, so we keep the intervals where the sign is negative.

Step 4 - Read off where the fraction is negative

That is negative two to two, and three to infinity.

Step 4 - Read off where the fraction is negative

Exclude x equals negative two and x equals two because the fraction is undefined there.

Step 4 - Read off where the fraction is negative

Exclude x equals three because the inequality is strict - less than, not less than or equal to.

Step 5 - State the solution and common pitfalls

The solution is the open interval from negative 2 to 2, union the open interval from 3 to infinity.

Step 5 - State the solution and common pitfalls

Two common mistakes: multiplying both sides by x squared minus four without checking its sign, which can flip the inequality.

Step 5 - State the solution and common pitfalls

And including the excluded values x equals negative two or x equals two in the answer.

Step 5 - State the solution and common pitfalls

Always use open brackets at values where the denominator is zero.