Solve a linear inequality
The answer is a set of values. A boundary separates values that satisfy the inequality from values that do not
- Use inverse operations Add or subtract the same amount on both sides. Positive multiplication and division preserve the order
- Reverse for a negative factor Multiplication or division by a negative number reverses the inequality sign \(\displaystyle \begin{aligned}-7x&>21\\x&<-3\end{aligned}\)
- Transfer equation methods Clear a known positive denominator, undo or expand a bracket, and collect unknown terms on both sides. Apply each operation to both sides and retain the negative-factor reversal rule
- Include equality only when allowed A strict sign excludes the boundary. An inclusive sign includes it
- Check the original inequality Test the boundary and a value on the proposed solution side. A single successful test checks your result; it does not replace the algebra
- Distinguish integers from real numbers For an integer answer, select the first or last allowed integer. A strict real-number upper bound has no greatest real solution \(\displaystyle x<\frac{19}{2}\)
- Cancellation cases After the variable terms cancel, a true comparison means all real values and a false comparison means no real values
- Contextual domain Apply whole-number and non-negative constraints separately from the algebraic inequality. Verify the selected integer and the next excluded one
Worked example
Solve the inequality
\[17-3x<8\]
- \(\displaystyle 17-3x-17<8-17\)Subtract the constant from both sides
- \(\displaystyle -3x<-9\)Simplify each side
- \(\displaystyle \frac{-3x}{-3}>\frac{-9}{-3}\)Divide by the negative coefficient and reverse the sign
- \(\displaystyle x>3\)Every value greater than the boundary is a solution; the boundary is excluded
- \(\displaystyle 17-3(3)=8\)At the boundary the sides are equal, so the strict inequality is false
- \(\displaystyle 17-3(4)=5<8\)A value on the proposed side satisfies the original inequality
Common mistakes
Keeping the sign after negative division
✕\(\displaystyle -8x<24\Rightarrow x<-3\)✓\(\displaystyle -8x<24\Rightarrow x>-3\)
Dividing both sides by a negative number reverses the orderDropping an allowed boundary
✕\(\displaystyle x\le-6\Rightarrow x<-6\)✓\(\displaystyle x\le-6\)
Replacing an inclusive sign with a strict sign removes a valid solutionChanging order after subtraction
✕\(\displaystyle x+9>14\Rightarrow x<5\)✓\(\displaystyle x+9>14\Rightarrow x>5\)
Subtracting the same number from both sides preserves the orderDifficulty levels
introSolve positive two-step inequalities. Enter an exact boundary and separately choose the complete relation symbol
standardSolve positive and negative two-step, whole-numerator fraction and bracket inequalities with exact boundaries, or find a requested integer threshold
challengeSolve one-variable inequalities with unknown terms on both sides. Coordinate-plane half planes are outside this placement
examChoose a complete one-variable inequality solution or the requested integer threshold, using the same operation and number bands
Questions about this topic
Does subtraction reverse the sign?+
No. Adding or subtracting the same number preserves the order. Negative multiplication or division reverses it
Is the boundary always a solution?+
Only an inclusive inequality allows equality at the boundary
Why might an integer answer differ from the boundary?+
The boundary may be excluded or may lie between integers. Choose an integer that belongs to the solution set
Should the sign reverse when I add a negative number?+
No. Adding or subtracting the same quantity preserves order. Multiplying or dividing by a negative number reverses it
What if all variable terms cancel?+
Evaluate the remaining comparison. If it is true, every real value is a solution; if false, no real value is a solution
Can I divide by the unknown?+
Only after considering its sign and excluding zero where division is used. A single unqualified division can lose cases or use the wrong comparison direction
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 10 Mathematics, chapter 8: Linear equations and inequalities
In this chapter
8A Linear equations and expressions: Solve a linear equationRecognise linear equations8A Linear equations and expressions: Set up and solve a practical linear equation8B Equations with brackets and fractions: Solve a linear equation8B Equations with brackets and fractions: Set up and solve a practical linear equationSolve equations that contain algebraic fractionsSubstitute values into a formulaSubstitute into and rearrange a formulaRead, draw and match inequality number lines8D Linear inequalities: Solve and interpret a linear inequationRead the point of intersection of two linesSolve a pair of simultaneous linear equationsSolve simultaneous equations by substitution8F Simultaneous equations by substitution: Solve a practical problem with simultaneous equations8F Simultaneous equations by substitution: Set up a pair of simultaneous equations from a description8F Simultaneous equations by substitution: Build and use a formula in two unknownsSolve simultaneous equations by elimination8G Simultaneous equations by elimination: Solve a practical problem with simultaneous equations8G Simultaneous equations by elimination: Set up a pair of simultaneous equations from a description8G Simultaneous equations by elimination: Build and use a formula in two unknowns8H Linear inequalities in the coordinate plane: Solve and interpret a linear inequation