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Geometry and Trigonometry

Geometry and Trigonometry

Introduction to geometry and trigonometry, exploring shapes, angles, and relationships forming the foundation for spatial reasoning and advanced mathematical problem-solving.

Geometry and Trigonometry

Introduction to geometry and trigonometry, exploring shapes, angles, and relationships forming the foundation for spatial reasoning and advanced mathematical problem-solving.

Linear Equations and Inequalities

4 min read

Concepts of Geometry

4 min read

Angle Relationships

3 min read

Congruence Proof

0:21

Congruence Proof Practice

3 tasks

Points, Lines, Planes

23 min read

Linear Equations and Inequalities

4 min read
Mar 31, 2026
Free

In this lesson

1. Linear EquationsStandard formExampleSolution steps2. Structure of Linear Expressions3. Solving Linear EquationsGeneral strategyExample4. Linear InequalitiesExample5. Solving InequalitiesExample6. Solution RepresentationNumber line visualization7. Graphical Interpretation8. Key PropertiesEquality vs Inequality rules9. Common Mistakes10. Concept Summary
You will learn
  • What linear equations and inequalities are
  • How to solve equations in one variable
  • Rules for manipulating inequalities
  • Representing solutions on number lines
  • Interpreting solutions graphically (Desmos)

1. Linear EquationsLink to 1-linear-equations

A linear equation is an equation where the highest power of the variable is 1.

Standard formLink to 1-linear-equations-standard-form

ax+b=0,a≠0ax + b = 0, \quad a \neq 0ax+

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b
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ExampleLink to 1-linear-equations-example

2x+3=112x + 3 = 112x+3=11

Solution stepsLink to 1-linear-equations-solution-steps

  1. Subtract 3 from both sides
  2. Divide by 2
2x=8⇒x=42x = 8 \Rightarrow x = 42x=8⇒x=4
Remember

Whatever operation you perform on one side of an equation, you must perform on the other side.

2. Structure of Linear ExpressionsLink to 2-structure-of-linear-expressions

A linear expression has the form:

ax+bax + bax+b
ComponentMeaning
aaaslope / coefficient
xxxvariable
bbbconstant term

3. Solving Linear EquationsLink to 3-solving-linear-equations

General strategyLink to 3-solving-linear-equations-general-strategy

Given:

ax+b=cax + b = cax+b=c

Steps:

  • Move constants to one side
  • Isolate variable term
  • Divide by coefficient

ExampleLink to 3-solving-linear-equations-example

5x−7=185x - 7 = 185x−7=185x=25⇒x=55x = 25 \Rightarrow x = 55x=25⇒x=5
Note

Check your solution by substitution into the original equation.

4. Linear InequalitiesLink to 4-linear-inequalities

A linear inequality compares expressions using:

  • <<<
  • >>>
  • ≤\le≤
  • ≥\ge≥

ExampleLink to 4-linear-inequalities-example

2x+1<92x + 1 < 92x+1<9

On the graph, this corresponds to all xxx values to the left of 444 (not included).

Edit Graph on Desmos

5. Solving InequalitiesLink to 5-solving-inequalities

Solve similarly to equations, with one key rule:

Warning

If you multiply or divide by a negative number, reverse the inequality sign.

ExampleLink to 5-solving-inequalities-example

−3x>6-3x > 6−3x>6

Divide by −3-3−3:

x<−2x < -2x<−2

6. Solution RepresentationLink to 6-solution-representation

Solutions of inequalities are often intervals.

InequalityInterval
x<2x < 2x<2(−∞,2)(-\infty, 2)(−∞,2)
x≤2x \le 2x≤2(−∞,2](-\infty, 2](−∞,2]

Number line visualizationLink to 6-solution-representation-number-line-visualization

IMAGE: Number line showing open circle at 2 for x<2x < 2x<2 and shaded left region

7. Graphical InterpretationLink to 7-graphical-interpretation

Linear inequalities can be visualized as regions on a graph.

DESMOS: y = 2x + 1 with shading for y < 2x + 1, showing half-plane region below the line

8. Key PropertiesLink to 8-key-properties

Equality vs Inequality rulesLink to 8-key-properties-equality-vs-inequality-rules

OperationEquationInequality
Add/subtract same valuepreservedpreserved
Multiply/divide positivepreservedpreserved
Multiply/divide negativepreservedsign flips

9. Common MistakesLink to 9-common-mistakes

Important
  • Forgetting to flip inequality when dividing by negative
  • Incorrect distribution of terms
  • Losing sign when moving terms across equality

10. Concept SummaryLink to 10-concept-summary

  • Linear equations have one variable with degree 1
  • Inequalities introduce range of solutions instead of a single value
  • Solution sets can be represented algebraically, numerically, and graphically
  • Transformation rules must preserve logical equivalence (or adjust inequality direction)