Linear Equations and Inequalities: A Complete Guide

Linear equations and inequalities are the foundation of algebra. Almost every later topic, from simultaneous equations to calculus, relies on your ability to rearrange and solve them accurately. This guide shows you a reliable method for each, explains the one rule that makes inequalities different, and works through examples step by step.

What Is a Linear Equation?

A linear equation in one variable can be written as

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

The variable appears only to the first power, so there is exactly one solution: \(x = -\frac{b}{a}\). Its graph, \(y = ax + b\), is a straight line, which is where the name "linear" comes from.

The Golden Rule

An equation is like a balance. Whatever you do to one side, you must do to the other. You may:

• add or subtract the same quantity on both sides,
• multiply or divide both sides by the same non-zero number.

A Step-by-Step Method for Linear Equations

1. Remove brackets by expanding.
2. Clear fractions by multiplying every term by the LCM of the denominators.
3. Collect variable terms on one side and constants on the other.
4. Divide by the coefficient of the variable.
5. Check by substituting back.

Worked Example 1: Brackets

Solve \(3(2x - 5) - 4(x - 1) = 7\).

Step 1: Expand the brackets:

$$ 6x - 15 - 4x + 4 = 7 $$

Step 2: Simplify the left side:

$$ 2x - 11 = 7 $$

Step 3: Add 11 to both sides:

$$ 2x = 18 $$

Step 4: Divide by 2:

$$ x = 9 $$

Check: \(3(18 - 5) - 4(9 - 1) = 3(13) - 4(8) = 39 - 32 = 7\). Correct.

Worked Example 2: Fractions

Solve \(\dfrac{x + 2}{3} - \dfrac{x - 1}{4} = 2\).

Step 1: The LCM of 3 and 4 is 12. Multiply every term by 12:

$$ 4(x + 2) - 3(x - 1) = 24 $$

Step 2: Expand. Note that \(-3 \times -1 = +3\):

$$ 4x + 8 - 3x + 3 = 24 $$

Step 3: Simplify:

$$ x + 11 = 24 $$

Step 4: Subtract 11:

$$ x = 13 $$

Check: \(\frac{15}{3} - \frac{12}{4} = 5 - 3 = 2\). Correct.

Special Cases

• No solution: If the variables cancel and you get a false statement, such as \(2x + 3 = 2x + 5\) giving \(3 = 5\), there is no solution.
• Infinitely many solutions: If you get a true statement, such as \(2(x + 1) = 2x + 2\) giving \(2 = 2\), every real number is a solution.

What Is a Linear Inequality?

A linear inequality uses one of these symbols instead of \(=\):

• \(<\) less than
• \(>\) greater than
• \(\leq\) less than or equal to
• \(\geq\) greater than or equal to

Instead of a single value, the solution is a range of values.

The One Rule That Makes Inequalities Different

You solve inequalities exactly like equations, with one crucial exception:

When you multiply or divide both sides by a negative number, reverse the inequality sign.

Why? Consider \(2 < 5\). Multiply both sides by \(-1\): \(-2\) and \(-5\). Since \(-2 > -5\), the sign must flip.

Worked Example 3: Reversing the Sign

Solve \(5 - 3x \geq 17\).

Step 1: Subtract 5 from both sides:

$$ -3x \geq 12 $$

Step 2: Divide by \(-3\) and reverse the sign:

$$ x \leq -4 $$

Check with a test value: Take \(x = -5\): \(5 - 3(-5) = 20 \geq 17\). True. Take \(x = 0\): \(5 \geq 17\). False, as expected.

In interval notation the solution is \((-\infty, -4]\). The square bracket shows that \(-4\) is included.

Worked Example 4: A Double Inequality

Solve \(-3 < 2x + 1 \leq 9\).

Work on all three parts at once.

Step 1: Subtract 1 from every part:

$$ -4 < 2x \leq 8 $$

Step 2: Divide every part by 2 (positive, so no sign change):

$$ -2 < x \leq 4 $$

In interval notation: \((-2, 4]\). If \(x\) must be an integer, the solutions are \(-1, 0, 1, 2, 3, 4\).

Showing Solutions on a Number Line

• Use an open circle for \(<\) or \(>\) (endpoint not included).
• Use a closed (filled) circle for \(\leq\) or \(\geq\) (endpoint included).
• Shade the direction of the solution set.

For \(-2 < x \leq 4\), draw an open circle at \(-2\), a filled circle at \(4\), and shade between them.

Common Mistakes

• Forgetting to flip the sign when dividing or multiplying by a negative number.
• Multiplying only some terms when clearing fractions. Every term, including those on the right side, must be multiplied.
• Sign errors with brackets. \(-3(x - 1) = -3x + 3\), not \(-3x - 3\).
• Mixing up open and closed circles on number lines.
• Not checking. A quick substitution catches most slips.

Key Takeaways

• Solve linear equations by expanding, clearing fractions, collecting terms and dividing.
• Treat inequalities like equations, but reverse the sign whenever you multiply or divide by a negative.
• Double inequalities are solved by applying each step to all three parts.
• Express inequality solutions using a number line or interval notation, and always test a value.

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