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Analyse signs and solve a quadratic inequality

Start a set of 8
Step 1 of 6
Profit rule
\(\displaystyle P(q)=-2(q-3)(q-9)\)
Positive profit
\(\displaystyle P(q)>0\)

A small business models its profit from subscriptions, where q is the number of hundreds of subscriptions. For which q-values is the profit positive?

Year 11, VCE Units 1 & 2, Chapter 3, 4 difficulty levels, No calculatorBuild a worksheet

Signs and quadratic inequalities

For a quadratic with real coefficients and a non-zero leading coefficient, use its real zeros to divide the number line into intervals. With two distinct real roots, the sign changes at each root. At a repeated root it does not change. If there are no real roots, the sign is the sign of a everywhere. Include a zero only for a non-strict inequality

\[f(x)=a(x-r_1)(x-r_2)\]
  1. Find the zeros Set the quadratic equal to zero to mark the boundary values on a number line \(\displaystyle f(x)=(x-2)(x-6)=0\Rightarrow x=2,6\)
  2. Test an interval Choose one value in each interval and evaluate the sign of the quadratic \(\displaystyle f(0)=(0-2)(0-6)=12>0\)
  3. Track the sign For an upward-opening quadratic with two simple roots, the signs are positive, negative, positive from left to right \(\displaystyle f(x):\quad +\text{ when }x<2,\ -\text{ when }2<x<6,\ +\text{ when }x>6\)
  4. Apply the endpoints Strict inequalities exclude zeros, while non-strict inequalities include them \(\displaystyle f(x)>0\Rightarrow x=2,6\text{ are excluded}\)

Worked example

Solve the inequality shown

\[-\left(x+1\right)\left(x-5\right)\ge0\]
  1. \(\displaystyle f(x)=-\left(x+1\right)\left(x-5\right)\)Let f be the expression on the left of the inequality
  2. \(\displaystyle x=-1\text{ or }x=5\)The zeros are the boundary values where either factor is zero
  3. \(\displaystyle f(-2)=-7,\quad f(0)=5,\quad f(6)=-7\)Testing one value in each interval gives negative, positive, negative signs
  4. \(\displaystyle -1\le x\le5\)The non-strict inequality selects the positive interval and includes both zeros

Common mistakes

Reversing between and outside
✕\(\displaystyle 2<x<6\)✓\(\displaystyle x<2\text{ or }x>6\)
An upward-opening quadratic is positive outside two simple roots and negative between them
Including strict endpoints
✕\(\displaystyle x\le2\text{ or }x\ge6\)✓\(\displaystyle x<2\text{ or }x>6\)
A strict inequality does not include the roots where the quadratic equals zero
Changing sign at a repeated root
✕\(\displaystyle -(x-3)^2\ge0\Rightarrow x\in\mathbb{R}\)✓\(\displaystyle -(x-3)^2\ge0\Rightarrow x=3\)
A squared factor is never negative, so its negative is at most zero and reaches zero only at the repeated root

Difficulty levels

introSolve a factorised inequality with integer roots by reading the sign intervals
standardSolve a quadratic inequality in general or factorised form and handle its endpoints
challengeTrack signs for two distinct roots, a repeated root or no real roots, including non-monic and rearranged inequalities
examPresent a complete interval solution with correct endpoint inclusion for a quadratic inequality

Questions about this topic

Why are the roots placed on a number line?+

The roots divide the number line into intervals. The quadratic has a constant sign within each interval, so one test value identifies that interval

When are the roots included in the answer?+

Include a root for ≥ or ≤ because the quadratic may equal zero. Exclude it for > or < because the quadratic must be strictly positive or negative

Does the sign always change at a root?+

The sign changes at a simple root. At a repeated root the factor is squared, so the sign stays the same on both sides

Can I use a calculator?+

No. This topic is practised without a calculator

Which year level and chapter is this?+

Year 11, VCE Units 1 & 2, Mathematical Methods, chapter 3: Quadratic functions

In this chapter

Factorise, solve and reconstruct a rational quadraticCalculate and interpret a quadratic discriminantTransform a quadratic and find its key featuresComplete the square and link equivalent formsDetermine a quadratic rule from exact informationSolve a quadratic exactly with the quadratic formulaConstruct a rational quadratic from a surd rootSolve an equation that reduces to quadratic formAnalyse signs and solve a quadratic inequalityIntersections of a line and a quadratic graphDetermine parameter values from a discriminant conditionFormulate and solve a quadratic problemAnalyse and optimise a quadratic model