Worked example · Easy
Find integer values
Write down the integer values satisfying \(-4<x\leq1\).
The values must be greater than \(-4\), so \(-4\) is not included. The value 1 is included.
\[\boxed{x=-3,-2,-1,0,1}\]Chapter 24
Inequalities
Learn to solve linear inequalities accurately, represent solutions on number lines, and interpret or construct shaded regions on coordinate graphs.
By the end of this chapter, you should be able to
- use the symbols \(<,\leq,>,\geq\) correctly;
- find integer values satisfying an inequality;
- solve linear and compound inequalities and represent their solutions on a number line;
- remember to reverse the inequality sign when multiplying or dividing by a negative number;
- identify whether a boundary line should be solid or dashed;
- shade the correct side of a straight-line boundary by testing a point;
- find or describe a region satisfying two or more inequalities.
1
Subtopic 1
Solving inequalities
Exam-ready method
Inclusive endpoints use filled circles; strict endpoints use open circles. Reverse the sign whenever multiplying or dividing by a negative number, including in compound inequalities.
An inequality compares values that are not necessarily equal. The four main symbols are:
| Symbol | Meaning | Example |
|---|---|---|
| \(<\) | less than | \(x<4\) |
| \(\leq\) | less than or equal to | \(x\leq4\) |
| \(>\) | greater than | \(x>4\) |
| \(\geq\) | greater than or equal to | \(x\geq4\) |
The one special rule
You normally solve an inequality in the same way as an equation. However:
\[\boxed{\text{Multiplying or dividing by a negative number reverses the inequality sign.}}\]For example, if \(-2x<-6\), dividing by \(-2\) gives \(x>3\).
Number lines
An open circle excludes the endpoint; a filled circle includes it. Draw the ray towards smaller values for “less than” or towards larger values for “greater than”.
Worked example · Easy
Solve a simple inequality
Solve \(3x+5<20\).
\[3x<15\]\[\boxed{x<5}\]
Worked example · Medium
Unknown on both sides
Solve \(5x-7\leq2x+8\).
\[3x-7\leq8\]\[3x\leq15\]\[\boxed{x\leq5}\]
Worked example · Medium
Reverse the inequality
Solve \(4-3x<13\).
\[-3x<9\]
Divide by \(-3\), so reverse the sign.
\[\boxed{x>-3}\]Worked example · Exam-style
A compound inequality
Solve \(7<3x+1\leq16\).
Subtract 1 from all three parts:
\[6<3x\leq15\]Divide all three parts by 3:
\[\boxed{2<x\leq5}\]Worked example · Exam-style
Form and solve an inequality
A rectangle has width \(a\) cm and length \((2a-3)\) cm. Its perimeter is greater than 24 cm but no more than 42 cm. Find the possible values of \(a\).
Perimeter:
\[2a+2(2a-3)=6a-6\]So:
\[24<6a-6\leq42\]\[30<6a\leq48\]\[\boxed{5<a\leq8}\]Number-line reminder
Use an open point for \(<\) or \(>\), and a filled/closed point for \(\leq\) or \(\geq\).
New worked example · Exam-style
Divide compound inequality by a negative
Solve \(-10<5-3x\leq8\).
Worked method — show your mathematical steps and check restrictions or units.
\[-15<-3x\leq3\]
\[\text{Divide by }-3\text{ and reverse both signs}\]
\[5>x\geq-1\]
\[\boxed{-1\leq x<5}\]
Try it yourself — from confidence to examination level
Attempt each question before opening its step-by-step solution.
1Simple inequality · Build confidence[2 marks]
Solve \(4x-7\geq9\).
Show Answer
Step-by-step solution
\[4x\geq16\]
\[\boxed{x\geq4}\]
2Flip the sign · Develop skill[3 marks]
Solve \(9-2x<1\).
Show Answer
Step-by-step solution
\[-2x<-8\]
\[\boxed{x>4}\]
3Whole numbers · Exam-style[4 marks]
List integer values satisfying \(-3<2x+1\leq7\).
Show Answer
Step-by-step solution
\[-4<2x\leq6\]
\[-2<x\leq3\]
\[\boxed{x=-1,0,1,2,3}\]
4Contextual bound · Exam challenge[5 marks]
A taxi fare is \(4+3k\) dollars for \(k\) kilometres. It must cost less than 34 dollars. What is the greatest whole number of kilometres?
Show Answer
Step-by-step solution
\[4+3k<34\Rightarrow k<10\]
\[\boxed{k=9\text{ kilometres}}\]
2
Subtopic 2
Showing regions on graphs
Exam-ready method
Draw the equality line first. Solid means inclusive; dashed means strict. Test a point away from the boundary to choose the correct side.
An inequality involving \(x\) and \(y\) describes a region of the coordinate plane. The corresponding equation gives the boundary line.
Strict inequality
For \(<\) or \(>\), draw the boundary as a dashed line because points on the line are not included.
Inclusive inequality
For \(\leq\) or \(\geq\), draw a solid line because points on the line are included.
How to choose the shaded side
- Draw the boundary equation.
- Choose a test point not on the line. \((0,0)\) is usually easiest.
- Substitute the test point into the inequality.
- If the statement is true, shade the side containing that point. If false, shade the opposite side.
Worked example · Easy
Horizontal boundary
Describe the region \(y\geq3\).
Draw the solid horizontal line \(y=3\). Since values of \(y\) must be at least 3, shade above the line.
Boundary: solid. Region: above the line.
Worked example · Medium
Use a test point
Which side of \(y=2x-1\) satisfies \(y<2x-1\)?
The boundary is dashed. Test \((0,0)\):
\[0<2(0)-1\]\[0<-1\]This is false, so shade the side not containing \((0,0)\).
Worked example · Medium
Read a shaded vertical region
A graph has a dashed vertical boundary \(x=4\) and the region to the left is shaded. Write the inequality.
Dashed means no equality. Left of \(x=4\) means smaller \(x\)-values.
\[\boxed{x<4}\]Worked example · Exam-style
Rearrange before drawing
Show the region satisfying \(2x+3y\leq12\).
The boundary is \(2x+3y=12\), a solid line. Useful intercepts are \((0,4)\) and \((6,0)\).
Test \((0,0)\):
\[2(0)+3(0)\leq12\]True, so shade the side containing the origin.
New worked example · Exam-style
Rearrange a sloping boundary
For \(2x+3y>12\), give the boundary equation and test whether \((0,0)\) is in the region.
Worked method — show your mathematical steps and check restrictions or units.
\[3y>12-2x\Rightarrow y>4-\frac23x\]
\[\text{Boundary: }y=4-\frac23x\text{ (dashed)}\]
\[2(0)+3(0)=0\not>12\]
\[\boxed{(0,0)\text{ lies outside the region}}\]
Try it yourself — from confidence to examination level
Attempt each question before opening its step-by-step solution.
1Simple vertical region · Build confidence[2 marks]
Write the inequality for the region to the right of the dashed line \(x=-2\).
Show Answer
Step-by-step solution
\[\text{Right means greater }x,\ \text{dashed means strict}\]
\[\boxed{x>-2}\]
2Horizontal boundary · Develop skill[3 marks]
Write the inequality for the region on or below the solid line \(y=4\).
Show Answer
Step-by-step solution
\[\text{Below means smaller }y,\ \text{solid includes equality}\]
\[\boxed{y\leq4}\]
3Test a point · Exam-style[4 marks]
For \(y<2x+1\), does \((1,2)\) lie in the region?
Show Answer
Step-by-step solution
\[2<2(1)+1=3\text{ is true}\]
\[\boxed{\text{Yes}}\]
4Gradient-intercept conversion · Exam challenge[5 marks]
Rewrite \(3x+2y\geq12\) in the form \(y\geq mx+c\).
Show Answer
Step-by-step solution
\[2y\geq12-3x\]
\[\boxed{y\geq6-\frac32x}\]
3
Subtopic 3
Representing regions satisfying more than one inequality
Exam-ready method
Each inequality represents one half-plane; the solution is the overlap satisfying them all. To describe a graph, read every boundary separately and use a point in the required region.
When several inequalities must all be true, their solution is the intersection of the allowed regions. A useful textbook technique is to shade the regions that do not satisfy each inequality; the required region is then the part left unshaded.
Worked example · Medium
Three inequalities together
Describe the region satisfying \(x\geq0\), \(y\geq0\) and \(x+2y\leq8\).
\(x\geq0\) keeps the region on or to the right of the \(y\)-axis.
\(y\geq0\) keeps the region on or above the \(x\)-axis.
\(x+2y\leq8\) keeps the region on the side of the line containing \((0,0)\).
The common triangular region is the solution.
From a shaded region back to inequalities
- Write the equation of each boundary line.
- Check whether each line is solid or dashed.
- Choose a point inside the required region.
- Substitute that point to determine whether each sign is \(<,\leq,>\) or \(\geq\).
Worked example · Exam-style
List inequalities from boundaries
A required region lies below the solid line \(y=5\), above the dashed line \(y=x-2\), and to the right of the solid line \(x=-1\). Write the three inequalities.
Below solid \(y=5\): \(y\leq5\).
Above dashed \(y=x-2\): \(y>x-2\).
Right of solid \(x=-1\): \(x\geq-1\).
\[\boxed{y\leq5,\quad y>x-2,\quad x\geq-1}\]Worked example · Exam-style
Find vertices of a feasible region
The inequalities are \(x\geq0\), \(y\geq0\), \(x+y\leq6\) and \(x\leq4\). State the vertices of the region.
Boundaries are the axes, \(x+y=6\), and \(x=4\).
Intersections in the allowed region:
\[(0,0),\quad(4,0),\quad(4,2),\quad(0,6)\]Vertices: \(\boxed{(0,0),(4,0),(4,2),(0,6)}\).
New worked example · Exam-style
Three inequalities and vertices
Find the vertices of the region satisfying \(x\geq0\), \(y\geq0\) and \(x+2y\leq10\).
Worked method — show your mathematical steps and check restrictions or units.
\[x+2y=10\text{ meets axes at }(10,0)\text{ and }(0,5)\]
\[\text{The first two inequalities restrict the region to quadrant I}\]
\[\boxed{(0,0),(10,0),(0,5)}\]
Try it yourself — from confidence to examination level
Attempt each question before opening its step-by-step solution.
1Check a coordinate · Build confidence[2 marks]
Does \((2,3)\) satisfy \(x\geq0\), \(y>0\), and \(x+y\leq6\)?
Show Answer
Step-by-step solution
\[2\geq0,\quad3>0,\quad2+3=5\leq6\]
\[\boxed{\text{Yes, all three hold}}\]
2Two sloping boundaries · Develop skill[3 marks]
For \(y\geq x\) and \(y\leq6-x\), find where their boundaries intersect.
Show Answer
Step-by-step solution
\[y=x\text{ and }y=6-x\]
\[2x=6\Rightarrow x=3,\quad y=3\]
\[\boxed{(3,3)}\]
3First-quadrant region · Exam-style[4 marks]
State the inequalities for the triangle in the first quadrant on or below \(2x+y=8\).
Show Answer
Step-by-step solution
\[\text{First quadrant: }x\geq0,\ y\geq0\]
\[\text{On or below the line: }2x+y\leq8\]
\[\boxed{x\geq0,\quad y\geq0,\quad2x+y\leq8}\]
4Region test · Exam challenge[5 marks]
Is \((3,1)\) inside the region \(x>0\), \(y\geq0\), \(3x+2y<12\)?
Show Answer
Step-by-step solution
\[3>0,\quad 1\geq0\]
\[3(3)+2(1)=11<12\]
\[\boxed{\text{Yes}}\]
Chapter 24 practice
Start with one-variable inequalities and number lines, then move to shaded graph regions and systems of inequalities. Keep each answer closed until you have attempted the question.
A. Build confidence
1Integer solutions[2 marks]
Write the integer values satisfying \(-2<x\leq3\).
Show Answer
Method: Check whether the inequality includes equality, apply reversible algebraic operations carefully, and test the resulting answer or boundary.
\[\boxed{x=-1,0,1,2,3}\]2Simple inequality[2 marks]
Solve \(4x+3<19\).
Show Answer
Method: Check whether the inequality includes equality, apply reversible algebraic operations carefully, and test the resulting answer or boundary.
\[4x<16\]\[\boxed{x<4}\]3Inclusive inequality[2 marks]
Solve \(7x-5\geq16\).
Show Answer
Method: Check whether the inequality includes equality, apply reversible algebraic operations carefully, and test the resulting answer or boundary.
\[7x\geq21\]\[\boxed{x\geq3}\]4Reverse the sign[3 marks]
Solve \(10-2x>18\).
Show Answer
Method: Check whether the inequality includes equality, apply reversible algebraic operations carefully, and test the resulting answer or boundary.
\[-2x>8\]Divide by \(-2\) and reverse the sign.
\[\boxed{x<-4}\]B. Developing skill
5Unknown on both sides[3 marks]
Solve \(6x-7\leq2x+9\).
Show Answer
Method: Check whether the inequality includes equality, apply reversible algebraic operations carefully, and test the resulting answer or boundary.
\[4x-7\leq9\]\[4x\leq16\]\[\boxed{x\leq4}\]6Bracket[3 marks]
Solve \(3(x-4)>2x+5\).
Show Answer
Method: Check whether the inequality includes equality, apply reversible algebraic operations carefully, and test the resulting answer or boundary.
\[3x-12>2x+5\]\[\boxed{x>17}\]7Compound inequality[3 marks]
Solve \(-5<2x+1\leq11\).
Show Answer
Method: Check whether the inequality includes equality, apply reversible algebraic operations carefully, and test the resulting answer or boundary.
\[-6<2x\leq10\]\[\boxed{-3<x\leq5}\]8Context[4 marks]
The perimeter of a square of side \(s\) cm is at least 28 cm but less than 52 cm. Find the possible values of \(s\).
Show Answer
Method: Check whether the inequality includes equality, apply reversible algebraic operations carefully, and test the resulting answer or boundary.
\[28\leq4s<52\]\[\boxed{7\leq s<13}\]C. Regions on graphs
9Boundary type[2 marks]
Should the boundary for \(y>3x-1\) be solid or dashed? Explain.
Show Answer
\(\boxed{\text{Dashed}}\), because \(>\) does not include points on the boundary line.
10Test a point[3 marks]
For \(y\leq x+4\), does the point \((0,0)\) lie in the required region?
Show Answer
Method: Check whether the inequality includes equality, apply reversible algebraic operations carefully, and test the resulting answer or boundary.
\[0\leq0+4\]This is true, so \(\boxed{(0,0)\text{ lies in the region}}\).
11Write the inequality[3 marks]
A graph has a solid horizontal boundary \(y=-2\), with the region above it shaded. Write the inequality.
Show Answer
Solid means equality is included; above means greater values of \(y\).
\[\boxed{y\geq-2}\]12Straight-line region[4 marks]
For \(3x+2y<12\), state the boundary line, whether it is solid or dashed, and whether \((0,0)\) is in the required region.
Show Answer
Boundary: \(3x+2y=12\).
It is dashed because the inequality is strict.
At \((0,0)\), \(0<12\), which is true, so the origin is in the required region.
D. Exam-style challenge
13Three inequalities[5 marks]
Describe the region satisfying \(x\geq0\), \(y\geq0\) and \(2x+y\leq8\). State the intercepts of the sloping boundary.
Show Answer
The region lies in the first quadrant and on/below the solid line \(2x+y=8\).
If \(x=0\), \(y=8\), so one intercept is \((0,8)\).
If \(y=0\), \(2x=8\), so \(x=4\), giving \((4,0)\).
14Identify boundaries[5 marks]
A region is below the solid line \(y=6\), above the dashed line \(y=2x-4\), and left of the solid line \(x=3\). Write the three inequalities.
Show Answer
Method: Check whether the inequality includes equality, apply reversible algebraic operations carefully, and test the resulting answer or boundary.
\[\boxed{y\leq6,\quad y>2x-4,\quad x\leq3}\]15Compound inequality and integers[5 marks]
Find all integer values of \(x\) satisfying \(-7\leq3x+2<11\).
Show Answer
Method: Check whether the inequality includes equality, apply reversible algebraic operations carefully, and test the resulting answer or boundary.
\[-9\leq3x<9\]\[-3\leq x<3\]Integers:
\[\boxed{-3,-2,-1,0,1,2}\]16Context and integer restriction[5 marks]
A school group has \(n\) students. A bus costs $420 plus $8 per student. The total cost must be more than $500 but no more than $620. Find the possible integer values of \(n\).
Show Answer
\[500<420+8n\leq620\]\[80<8n\leq200\]\[10<n\leq25\]
Since \(n\) is an integer:
\[\boxed{n=11,12,\ldots,25}\]17Vertices[6 marks]
The region satisfies \(x\geq0\), \(y\geq0\), \(x\leq5\) and \(x+y\leq7\). State its vertices.
Show Answer
Starting at the origin and following the boundaries:
\[(0,0),\ (5,0),\ (5,2),\ (0,7)\]So the vertices are \(\boxed{(0,0),(5,0),(5,2),(0,7)}\).
18Check a candidate point[4 marks]
Does \((2,3)\) satisfy all three inequalities \(x>0\), \(y\leq4\), and \(x+2y<9\)?
Show Answer
\(2>0\): true.
\(3\leq4\): true.
\(2+2(3)=8<9\): true.
So \(\boxed{(2,3)\text{ satisfies all three inequalities}}\).
Chapter 24 high-grade checklist
Inequality symbols
I know when equality is included and can read compound inequalities correctly.
Negative rule
I always reverse the inequality sign when multiplying or dividing by a negative number.
Number lines
I use open and closed endpoints correctly and shade in the correct direction.
Graph boundaries
I use dashed lines for strict inequalities and solid lines for inclusive inequalities.
Test points
I can test a convenient point to determine which side of a boundary is required.
Multiple inequalities
I can identify the common region satisfying all inequalities and work backwards from a region to its inequalities.