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Algebra 1 Unit 2: Linear equations & inequalities

Solving an equation means finding the value that makes it true. Keep the equation balanced by doing the same thing to both sides.

Covers: One- and multi-step equations, variables on both sides, literal equations, inequalities and their graphs

Study guide

Solving an equation means isolating the variable using inverse operations, doing the same thing to both sides to keep them equal. One-step equations need a single inverse operation; multi-step equations usually go: distribute, combine like terms on each side, move variable terms to one side, move constants to the other, then divide.

When variables appear on both sides, subtract one variable term from both sides first. If the variables cancel and leave a true statement, every number is a solution; if they leave a false statement, there is no solution. Always check by substituting your answer into the original equation.

Literal equations contain several variables, like formulas. Solving A = lw for w means treating the other letters as numbers: divide both sides by l to get w = A/l.

Inequalities use , ≤ and ≥, and their solutions are ranges of numbers. Solve them like equations, with one key rule: multiplying or dividing both sides by a negative number reverses the inequality sign. Graph solutions on a number line with an open circle for (the endpoint isn't included) and a closed circle for ≤ or ≥ (it is), shading the direction of the solutions. Compound inequalities joined by "and" are the overlap; "or" is either piece.

"At least" and "no less than" mean ≥, "at most" and "no more than" mean ≤, "more than" or "exceeds" means >, and "fewer than" means <. Decide which side holds the variable quantity, then write the inequality before solving. To convert units, multiply by conversion fractions such as 3 ft/1 yd, arranged so the unwanted unit cancels top and bottom. Chain several fractions for multi-step conversions, and write the units at every step to check that only the target unit remains.

Key ideas

• Undo addition/subtraction first, then multiplication/division.
• Collect variable terms on one side.
• Literal equations: solve for one letter in terms of the others.
• Inequalities work the same, except multiplying or dividing by a negative flips the sign.
• Graph inequalities with open circles for and closed circles for ≤ or ≥.

Formulas and rules

Common mistakes

• Forgetting to flip the inequality sign.
• Doing an operation to only one side.

Words to know

Isolating the variable

Getting the variable alone on one side; the goal of solving.

Open vs closed circle on a number line

Open: . Closed: ≤ or ≥.

Literal equation

An equation with several variables, solved for one of them (A = lw → w = A/l).

Flipping an inequality

Required when multiplying or dividing both sides by a negative.

Checking a solution

Substituting it back into the original equation.

Equation

A statement that two expressions are equal.

Multi-step equation

Distribute, combine like terms, move variables to one side, then divide.

Identity (equation)

An equation true for every value, like 2x + 4 = 2(x + 2).

Compound inequality

Two inequalities joined by "and" (overlap) or "or" (either).

Solution set

All the values that make an equation or inequality true.

Graphing an inequality

Open circle for , closed circle for ≤ or ≥, shade the solutions.

Sample practice questions

35 more questions like these, from level 1 to 5, are in the practice set.