3.1 Linear Equations & Inequalities

Solving Multi-Step Linear Equations

A linear equation has a variable raised only to the first power, such as 3x + 7 = 22. To solve one means to find the value of the variable that makes both sides equal. The strategy is always the same: simplify each side, then undo operations in reverse order using inverse operations (addition undoes subtraction, multiplication undoes division).

The standard order is:

1. Distribute to clear parentheses.
2. Combine like terms on each side.
3. Move all variable terms to one side, all constants to the other.
4. Divide by the coefficient of the variable.
5. Check by substituting back.

Worked example: Solve 5(x − 3) + 4 = 2x + 7.

• Distribute: 5x − 15 + 4 = 2x + 7
• Combine like terms: 5x − 11 = 2x + 7
• Subtract 2x from both sides: 3x − 11 = 7
• Add 11 to both sides: 3x = 18
• Divide by 3: x = 6
• Check: 5(6 − 3) + 4 = 19 and 2(6) + 7 = 19. ✓

Equations with Fractions

Fractions are easiest to remove first. Find the least common denominator (LCD) of every fraction, then multiply EVERY term on both sides by that LCD. This produces an equivalent equation with whole-number coefficients.

Worked example: Solve (x/3) + (1/2) = (5/6).

• The denominators are 3, 2, and 6, so the LCD is 6.
• Multiply every term by 6: 6·(x/3) + 6·(1/2) = 6·(5/6)
• Simplify: 2x + 3 = 5
• Subtract 3: 2x = 2, so x = 1.
• Check: (1/3) + (1/2) = (2/6) + (3/6) = (5/6). ✓

Literal Equations

A literal equation contains several letters and is solved for one of them in terms of the others — exactly what you do when rearranging a formula. Treat the target variable as the unknown and every other letter as a number.

Worked example: Solve the area formula A = (1/2)bh for h.

• Multiply both sides by 2: 2A = bh
• Divide both sides by b: h = 2A / b

Worked example: Solve y = mx + b for x.

• Subtract b: y − b = mx
• Divide by m: x = (y − b) / m

Linear Inequalities and the Sign-Flip Rule

An inequality uses <, >, ≤, or ≥ instead of an equals sign and has a whole RANGE of solutions. You solve it just like an equation with one critical exception: whenever you multiply or divide both sides by a negative number, you must reverse (flip) the inequality symbol. This is the single most common ALEKS trap.

Worked example: Solve −2x + 5 < 11.

• Subtract 5: −2x < 6
• Divide by −2 and FLIP the sign: x > −3

Graph this on a number line with an open circle at −3 (because > is strict) and shading to the right. Use a closed/filled circle for ≤ or ≥.

Compound Inequalities

A compound inequality joins two inequalities. An 'and' statement (also written as a three-part inequality) requires BOTH to be true — the solution is the overlap. An 'or' statement requires at least one — the solution is the union.

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

• Subtract 3 from all three parts: −4 ≤ 2x < 6
• Divide all three parts by 2: −2 ≤ x < 3

The solution is every number from −2 (included) up to 3 (not included).

Common Mistakes

• Forgetting to flip the sign when dividing/multiplying by a negative — the #1 error.
• Multiplying only SOME terms by the LCD instead of every term.
• Distributing a negative incorrectly: −(x − 4) = −x + 4, not −x − 4.
• Using an open circle when the symbol is ≤ or ≥ (it should be filled).

Equations With Variables on Both Sides

Many ALEKS items place variable terms on both sides of the equation. The plan is to collect all variable terms on one side and all constants on the other, choosing the side that keeps the variable coefficient positive when convenient.

Worked example: Solve 7x − 4 = 3x + 12.

• Subtract 3x from both sides: 4x − 4 = 12.
• Add 4 to both sides: 4x = 16.
• Divide by 4: x = 4.
• Check: 7(4) − 4 = 24 and 3(4) + 12 = 24. ✓

A few equations have no solution or are an identity. If solving produces a false statement like 5 = 8, there is no solution. If it produces a true statement like 6 = 6, every real number works (infinitely many solutions).

Worked example: Solve 2(x + 3) = 2x + 6. Distributing gives 2x + 6 = 2x + 6; subtracting 2x leaves 6 = 6, a true statement, so the solution is all real numbers.

Writing Inequalities From Words

Placement problems often translate a sentence into an inequality. Phrases such as "at least" mean ≥, "at most" means ≤, "more than" means >, and "less than" means <.

Worked example: A taxi charges a 3-dollar base fee plus 2 dollars per mile, and a rider has at most 25 dollars. How many miles m can they travel? Set up 3 + 2m ≤ 25. Subtract 3: 2m ≤ 22. Divide by 2: m ≤ 11 miles. Because dividing by a positive number, the sign does not flip.