Linear Equations and Inequalities

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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

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

ExampleLink to 1-linear-equations-example

$$2x + 3 = 11$$

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

1. Subtract 3 from both sides
2. Divide by 2

$$2x = 8 \Rightarrow x = 4$$

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

A linear expression has the form:

$$ax + b$$

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

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

Given:

$$ax + b = c$$

Steps:

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

ExampleLink to 3-solving-linear-equations-example

$$5x - 7 = 18$$ $$5x = 25 \Rightarrow x = 5$$

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4. Linear InequalitiesLink to 4-linear-inequalities

A linear inequality compares expressions using:

• $<$
• $>$
• $\le$
• $\ge$

ExampleLink to 4-linear-inequalities-example

$$2x + 1 < 9$$

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5. Solving InequalitiesLink to 5-solving-inequalities

Solve similarly to equations, with one key rule:

ExampleLink to 5-solving-inequalities-example

$$-3x > 6$$

Divide by $-3$:

$$x < -2$$

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6. Solution RepresentationLink to 6-solution-representation

Solutions of inequalities are often intervals.

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

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

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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

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8. Key PropertiesLink to 8-key-properties

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

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9. Common MistakesLink to 9-common-mistakes

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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)