2.2 Linear Equations & Inequalities
Key Takeaways
- Solve linear equations by clearing fractions, distributing, collecting variables on one side, and isolating the variable with inverse operations.
- When solving inequalities, multiplying or dividing by a negative number reverses the inequality symbol.
- Literal equations are rearranged for a chosen variable using the same inverse-operation steps as numeric equations.
- Word problems become equations once you define the variable explicitly and translate each phrase into algebra.
- Number-line graphs use an open circle for strict inequalities and a closed circle for inclusive ones.
Solving Multi-Step Linear Equations
A linear equation has the variable to the first power only. The goal is to isolate the variable using inverse operations, keeping both sides balanced. A reliable order is: clear fractions, distribute, combine like terms on each side, move variables to one side, then undo addition and multiplication.
Variables on both sides: Solve 3(x - 2) = 5x + 4. Distribute: 3x - 6 = 5x + 4. Subtract 3x from both sides: -6 = 2x + 4. Subtract 4: -10 = 2x. Divide by 2: x = -5.
Check by substituting x = -5: left side 3(-5 - 2) = -21; right side 5(-5) + 4 = -21. The sides match, so the solution is verified.
Two special outcomes can occur. If the variable cancels and leaves a true statement like 8 = 8, every number works and the equation has infinitely many solutions (an identity). If it leaves a false statement like 3 = 7, no value works and there is no solution. Recognizing these saves time and earns interpretation credit on the Regents.
Equations with Fractions
Clear fractions first by multiplying every term by the least common denominator (LCD). This avoids fraction arithmetic mid-problem.
Worked example: Solve x/3 + 1/2 = x/6 - 1. The LCD of 3, 2, and 6 is 6. Multiply each term by 6: 6 times x/3 = 2x, 6 times 1/2 = 3, 6 times x/6 = x, 6 times -1 = -6. The equation becomes 2x + 3 = x - 6. Subtract x: x + 3 = -6. Subtract 3: x = -9.
Multiplying every term, including the constants, by the LCD is the step students miss most often.
Proportions are a special fractional equation solved by cross-multiplying. To solve (x + 1)/4 = 3/2, cross-multiply to get 2(x + 1) = 12, then x + 1 = 6 and x = 5. Cross-multiplication is just clearing denominators in one step and works only when each side is a single fraction.
Literal Equations and Word Problems
A literal equation is solved for one variable in terms of the others using the same inverse-operation logic.
Worked example: Solve A = (1/2)bh for h. Multiply both sides by 2: 2A = bh. Divide both sides by b: h = 2A / b.
For word problems, always define the variable, then translate phrase by phrase. "A gym charges a 30 dollar joining fee plus 20 dollars per month. After how many months does the total reach 150 dollars?" Let m be months. Total cost: 30 + 20m = 150. Subtract 30: 20m = 120. Divide: m = 6 months. Writing the equation before testing answers prevents misreading "plus" as multiplication or reversing the fee and the rate.
Literal rearrangement shows up in science-flavored Regents items too. To solve P = 2L + 2W for W, subtract 2L: P - 2L = 2W, then divide by 2: W = (P - 2L) / 2. The variable you solve for should end up alone on one side, and every other letter is treated as a constant during the steps.
Solving and Graphing Inequalities
Inequalities are solved like equations with one extra rule: multiplying or dividing both sides by a negative number reverses the inequality symbol.
Worked example: Solve -2x + 7 > 1. Subtract 7: -2x > -6. Divide by -2 and flip the sign: x < 3.
Graph the solution on a number line using these conventions:
| Symbol | Circle | Direction |
|---|---|---|
| > or < | Open circle | Shade away from the point |
| >= or <= | Closed (filled) circle | Shade away, including the point |
For x < 3, place an open circle at 3 and shade to the left. Use a closed circle only when the value itself is included (>= or <=).
Compound Inequalities and Interval Reasoning
A compound inequality joins two conditions. An "and" inequality such as -1 <= x < 4 means x is at least -1 and less than 4; it graphs as the segment between them, closed at -1 and open at 4. An "or" inequality such as x < -2 or x > 5 graphs as two separate rays.
Worked example: Solve -3 < 2x + 1 <= 7. Subtract 1 from all three parts: -4 < 2x <= 6. Divide all three parts by 2: -2 < x <= 3. Because you divided by a positive 2, no signs flip.
In modeling contexts, the interval describes a feasible range. If a delivery van must carry between 200 and 500 pounds, the constraint is 200 <= w <= 500, an inclusive interval at both ends.
Inequalities also answer "how many" questions. "A worker earns 15 dollars per hour and needs at least 600 dollars" becomes 15h >= 600, so h >= 40 hours. Because hours cannot be negative or fractional in many contexts, the realistic solution is a whole number 40 or greater. Always check whether the context limits the interval to non-negative or integer values before reporting the answer, since the Regents expects the solution to make sense for the situation described.
Solve for x: 4x - 9 = 2x + 7.
What is the solution to the inequality -3x + 2 >= 14, and how is it graphed?