1.3 Algebra I: Linear Equations & Inequalities

Algebra converts word problems and technical formulas into solvable equations — a skill tested directly on the NAPT and used implicitly in nearly every physics and chemistry calculation later in this guide. Rearranging Ohm's law, gas-law formulas, and stoichiometry ratios all rely on the same equation-solving steps taught here. This section covers solving multi-step linear equations, solving inequalities, and solving systems of two linear equations.

Solving Multi-Step Linear Equations

A linear equation is an equation where the variable appears only to the first power (no exponents, no square roots of the variable). Solving one means isolating the variable on one side using inverse operations, applied in a consistent order:

1. Distribute any parentheses.
2. Combine like terms on each side.
3. Move variable terms to one side and constants to the other (by adding or subtracting).
4. Divide by the coefficient of the variable to isolate it.

Worked example: Solve 4(x − 3) + 5 = 2x + 9.

• Distribute: 4x − 12 + 5 = 2x + 9
• Combine like terms: 4x − 7 = 2x + 9
• Move variable terms: 4x − 2x = 9 + 7 → 2x = 16
• Divide: x = 8

Check by substituting back into the ORIGINAL equation: 4(8 − 3) + 5 = 4(5) + 5 = 25, and 2(8) + 9 = 25. Both sides match, confirming x = 8.

Common Errors

• Forgetting to distribute to every term inside the parentheses — a frequent source of a wrong answer.
• Dropping a sign when moving a term across the equals sign (a term that was +5 becomes −5 on the other side, not staying positive).

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Solving Linear Inequalities

An inequality compares two expressions using <, >, ≤, or ≥ instead of an equals sign. You solve inequalities using the same steps as equations, with one critical exception:

When you multiply or divide both sides of an inequality by a negative number, you must flip the inequality symbol.

This single rule is one of the most frequently tested algebra traps on standardized STEM tests.

Worked example (flip required): Solve −2x + 5 > 13.

• Subtract 5 from both sides: −2x > 8
• Divide both sides by −2, and flip the inequality: x < −4

If you forget to flip the symbol, you would incorrectly report x > −4 — a mistake that reverses the entire solution set.

Worked example (no flip needed): Solve 3x + 7 ≤ 22.

• Subtract 7: 3x ≤ 15
• Divide by 3 (a positive number, so no flip): x ≤ 5

Reading the Solution

An inequality's solution is a range of values, not a single number. "x < −4" means every number less than −4 makes the original inequality true. You can check this with a test value: at x = −5, −2(−5) + 5 = 15, and 15 > 13 is true, confirming −5 is part of the solution set.

Compound Inequalities

A compound inequality combines two conditions into a single statement, such as −3 < 2x + 1 ≤ 9. Solve it by applying the same operation to all three parts at once.

Worked example: Solve −3 < 2x + 1 ≤ 9.

• Subtract 1 from all three parts: −4 < 2x ≤ 8
• Divide all three parts by 2 (positive, so no flip): −2 < x ≤ 4

The solution is every x greater than −2 and less than or equal to 4. If the operation had involved dividing by a negative number, all three parts — including the direction of both inequality symbols — would need to flip.

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Systems of Two Linear Equations

A system of equations is a set of two or more equations solved together, where the solution must satisfy every equation simultaneously. For two linear equations in two variables, the solution is the single (x, y) point where both lines intersect. Two efficient solving methods are shown below.

Worked example (elimination): Solve the system:

2x + y = 11 x − y = 1

The y-terms have opposite signs, so add the two equations directly:

(2x + y) + (x − y) = 11 + 1

3x = 12

x = 4

Substitute x = 4 back into the second equation: 4 − y = 1, so y = 3.

Solution: (x, y) = (4, 3).

Check both equations: 2(4) + 3 = 11 ✓ and 4 − 3 = 1 ✓.

Worked example (translating a word problem): Two numbers have a sum of 20 and a difference of 4. What are the two numbers?

Let x be the larger number and y be the smaller number, and translate the two sentences directly into equations:

x + y = 20 x − y = 4

Add the equations to eliminate y: 2x = 24, so x = 12. Substitute back into the first equation: 12 + y = 20, so y = 8.

The two numbers are 12 and 8. Check: 12 + 8 = 20 ✓ and 12 − 8 = 4 ✓.

Why This Matters for the NAPT

Word problems involving two unknowns — two numbers with a given sum and difference, or two rates that must both be satisfied — are almost always systems of equations in disguise. Recognizing the two relationships and translating them into two equations is the hard part; solving them uses the same elimination or substitution steps shown above.

Test-Day Tip

Always check your solution by substituting it back into both original equations. A value that satisfies only one equation is not the answer to the system — this is a common trap built into multiple-choice options.