Learning Objectives
  • Interpret and represent inequalities using symbols and number lines.
  • Solve linear and compound inequalities in one variable.
  • Reverse the inequality sign correctly when multiplying or dividing by a negative number.
  • Represent inequalities in two variables using boundary lines and regions.
  • Write inequalities that define a given region.
Key Terms
Strict inequality
An inequality using \lt or \gt , which excludes the boundary.
Inclusive inequality
An inequality using \le or \ge, which includes the boundary.
Compound inequality
Two inequalities joined to restrict a variable to an interval.
Boundary line
The line separating regions satisfying and not satisfying a two-variable inequality.
Feasible region
The region satisfying all listed inequalities; linear programming is not required here.
Integer solution
A whole-number value satisfying the inequality.
8th Edition Chapter Map
  • Inequality notation and number lines
  • Solving one-step and multi-step inequalities
  • Compound inequalities and integer solutions
  • Graphical inequalities in two variables
  • Listing inequalities from a region
Reading Inequality Symbols
Symbol Meaning Example
x\lt 4 x is less than 4 4 is excluded.
x\le4 x is at most 4 4 is included.
x\gt 4 x is greater than 4 4 is excluded.
x\ge4 x is at least 4 4 is included.

On a number line, use an open circle for a strict inequality and a closed circle for an inclusive inequality. Shade or draw an arrow in the direction of all permitted values.

Solving Like An Equation

Addition and subtraction can be performed on both sides without changing the inequality direction. Multiplying or dividing both sides by a positive number also preserves the direction.

Worked Example: Multi-Step Inequality

Question: Solve 5x-7\lt 18.

  1. Add 7: 5x\lt 25.
  2. Divide by 5: x\lt 5.
  3. Use an open circle at 5 and shade to the left.

Answer: x\lt 5.

Why The Sign Reverses For A Negative Multiplier

If 3\lt 7, multiplying both sides by -1 gives -3\gt -7. The order reverses on the number line. Therefore, whenever both sides are multiplied or divided by a negative number, reverse the inequality sign.

Worked Example: Negative Coefficient

Question: Solve 12-3x\ge27.

  1. Subtract 12: -3x\ge15.
  2. Divide by -3 and reverse the sign.
  3. x\le-5.

Answer: x\le-5.

Variables On Both Sides
Worked Example: Collect The Variable

Question: Solve 7x+4\gt 3x+20.

  1. Subtract 3x: 4x+4\gt 20.
  2. Subtract 4: 4x\gt 16.
  3. Divide by 4.

Answer: x\gt 4.

Compound Inequalities

A compound inequality such as -3\le2x-1\lt 7 can be solved by applying the same operation to all three parts.

Worked Example: Three-Part Inequality

Question: Solve -3\le2x-1\lt 7.

  1. Add 1 to every part: -2\le2x\lt 8.
  2. Divide every part by 2: -1\le x\lt 4.

Answer: -1\le x\lt 4.

Integer Solutions

If -2.4\lt x\le3.1 and x is an integer, then x=-2,-1,0,1,2,3. Do not list non-integers. If the question asks for the greatest or least integer solution, read the boundary carefully.

Two-Variable Inequalities

Replace the inequality by equality to draw its boundary. Use a broken line for \lt or \gt , and a solid line for \le or \ge. Test a point not on the line, often (0,0), to decide which side satisfies the inequality.

Worked Example: Test A Region

Question: Represent y\gt 2x-1.

  1. Draw the broken boundary line y=2x-1.
  2. Test (0,0): 0\gt -1 is true.
  3. The side containing the origin is the wanted region. If following Cambridge convention, shade the unwanted side unless the question instructs otherwise.

Answer: The solution is the region above the broken line y=2x-1.

Vertical And Horizontal Boundaries

x\lt 3 has a vertical boundary x=3 and the solution lies to the left. y\ge-2 has a horizontal solid boundary y=-2 and the solution lies above it.

Writing Inequalities From A Given Region

Identify each boundary line, decide whether it is included, and choose the correct side. A solid boundary means \le or \ge; a broken boundary means \lt or \gt . Test a point inside the required region to determine the direction.

Examination Guidance
  • When multiplying or dividing by a negative number, write the reversed sign immediately.
  • Use open and closed circles correctly on number lines.
  • For graphical inequalities, state whether the boundary is solid or broken.
  • Cambridge may use shading for unwanted regions; read the instruction before shading.
Common Mistakes
  • Treating an inequality exactly like an equation after division by a negative number.
  • Using a closed circle for \lt or \gt .
  • Drawing a solid boundary for a strict two-variable inequality.
  • Testing a point that lies on the boundary line.
Knowledge Check And Practice

1. Solve 4x+1\le17.

Answer: x\le4.

2. Solve 5-2x\gt 11.

Answer: -2x\gt 6, so x\lt -3.

3. Solve -5\lt 3x+1\le13.

Answer: -2\lt x\le4.

4. List the integer solutions of -1.5\le x\lt 3.2.

Answer: -1,0,1,2,3.

5. What boundary is used for y\le4x+2?

Answer: The solid line y=4x+2.
Extended Worked Practice
Solving A Compound Inequality

Question: Solve -5<2x+1\le 11.

  1. Subtract 1 from all three parts: -6<2x\le10.
  2. Divide all three parts by 2: -3<x\le5.

Answer: -3<x\le5. On a number line, use an open circle at -3 and a filled circle at 5.

Integer Solutions

Question: List the integer solutions of 4-3x<13 with x\le2.

  1. -3x<9.
  2. Divide by -3 and reverse the sign: x>-3.
  3. Combine with x\le2.

Answer: x=-2,-1,0,1,2.

Describing A Shaded Region

Question: A region lies above y=x-1, below y=5, and to the right of x=0. Write its inequalities.

Answer: y\ge x-1, y\le5 and x\ge0. A solid boundary is used because equality is included.

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