Substitution and the Test-Point Method
Substitution and the Test-Point Method
Substitution in Inequalities
After solving an inequality in a substituted variable, apply the range of that variable before converting the answer back to \(x\). In the following two tables assume \(a>0\) and \(b>0\).
Case 1: \(u<-a\) or \(u>b\)
| Substitution | After substitution | Result in \(x\) |
|---|---|---|
| \(u=e^x\) | \(e^x<-a\) or \(e^x>b\) | \(e^x<-a\) is rejected because \(e^x>0\). Hence \(e^x>b\), so \(x>\ln b\). |
| \(u=\ln x\) | \(\ln x<-a\) or \(\ln x>b\) | \(0<x<e^{-a}\) or \(x>e^b\), with \(x>0\). |
| \(u=|x|\) | \(|x|<-a\) or \(|x|>b\) | \(|x|<-a\) is rejected because \(|x|\ge0\). Hence \(x>b\) or \(x<-b\). |
Case 2: \(-a<u<b\)
| Substitution | After substitution | Result in \(x\) |
|---|---|---|
| \(u=e^x\) | \(-a<e^x<b\) | Since \(e^x>0\), this becomes \(0<e^x<b\), so \(x<\ln b\). |
| \(u=\ln x\) | \(-a<\ln x<b\) | \(e^{-a}<x<e^b\). |
| \(u=|x|\) | \(-a<|x|<b\) | Since \(|x|\ge0\), this becomes \(0\le|x|<b\), then \(|x|<b\), so \(-b<x<b\). |
Test-Point Method
- Move all terms to one side so the other side is \(0\).
- Factorise the expression fully.
- Mark the critical points on a number line. Exclude any value that makes a denominator zero.
- Choose a test point in each interval to determine the sign of the expression, then select the interval or intervals that satisfy the inequality.
Important Notes
- An alternative method is to sketch the corresponding graph and inspect where it lies above or below the \(x\)-axis.
- For a rational inequality, do not cross-multiply unless the sign of the denominator is known.
- Do not multiply an inequality by an expression unless its sign is known.
- After cancelling common numerator and denominator factors, a zero or pole of even multiplicity does not change the sign; one of odd multiplicity changes the sign. A repeated root can have either parity. Retain every exclusion from the original denominator, including cancelled factors.
- Denominator roots are excluded even when the inequality is non-strict.
Sign Charts
| Expression | Test-point sign chart | Solution |
|---|---|---|
| \((2x-5)(x-3)(x+4)\le0\) | \(x\le-4\) or \(\dfrac52\le x\le3\) | |
| \((2x-5)(x-3)^2(x+4)\le0\) | \(-4\le x\le\dfrac52\) or \(x=3\) | |
| \(\dfrac{(x+2)(3x+2)}{x(x-1)}\le0\) | \(-2\le x\le-\dfrac23\) or \(0<x<1\) | |
| \(\dfrac{x^2-2}{x^2(x-1)}>0\) | \(-\sqrt2<x<0\) or \(0<x<1\) or \(x>\sqrt2\) |
Sign patterns around roots and excluded values
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