2.2 Equations, Inequalities, and Systems

ACT Algebra Is About Equivalent Moves

ACT's detailed math description names Algebra as a 17-20% reporting area and describes solving, graphing, and modeling several types of equations. The fastest students are not the ones who perform the most steps; they are the ones who make legal moves and avoid creating extra work. Every equation move should preserve the same solution set unless the problem deliberately asks for a transformation.

Start simple. Clear parentheses, combine like terms, move variable terms to one side, move constants to the other side, then divide by the coefficient. That order prevents sign mistakes. When fractions appear, multiply both sides by the least common denominator early, but distribute carefully after doing so.

Equation Type Decision Table

Linear Equations and Formula Rearranging

A linear equation has variables to the first power. For 5(2x - 3) = 4x + 21, distribute first: 10x - 15 = 4x + 21. Move terms: 6x = 36, so x = 6. If you instead add 15 only to one part of the right side, the equation stops being equivalent.

Formula rearranging is the same skill with letters. If A = P(1 + rt) and the question asks for r, divide by P: A/P = 1 + rt. Subtract 1: A/P - 1 = rt. Divide by t: r = (A/P - 1)/t. Do not plug in numbers until the formula shape is clear unless the problem supplies all values.

Inequalities

Inequalities follow the same arithmetic moves as equations, with one major rule: multiplying or dividing both sides by a negative number reverses the inequality direction. Adding, subtracting, and multiplying by a positive number do not reverse it.

For -3x + 7 <= 22, subtract 7 to get -3x <= 15. Divide by -3 and reverse the sign: x >= -5. A common wrong answer is x <= -5, which comes from forgetting that division by a negative changes the comparison.

Compound inequalities need both boundaries maintained. If -2 < 3x + 4 <= 16, subtract 4 from all three parts: -6 < 3x <= 12. Divide all three parts by 3: -2 < x <= 4. Keep the open and closed endpoints exactly as they appear after legal moves.

Absolute Value and Distance

Absolute value equations are distance questions. If |x - 5| = 9, then x - 5 = 9 or x - 5 = -9, so x = 14 or x = -4. If |x - 5| = -9, there is no solution because a distance cannot equal a negative number.

For inequalities, think near and far. |x - 5| < 9 means x is within 9 units of 5, so -4 < x < 14. |x - 5| > 9 means x is farther than 9 units from 5, so x < -4 or x > 14. A quick number-line sketch can prevent mixing these cases.

Systems: Substitute, Eliminate, or Interpret

A system of equations asks for values that make all equations true at the same time. Substitution is efficient when one equation already says x = something or y = something. Elimination is efficient when adding or subtracting equations can cancel a variable.

Example: solve 2x + y = 17 and x - y = 4. Add the equations to eliminate y: 3x = 21, so x = 7. Substitute into x - y = 4: 7 - y = 4, so y = 3. If the question asks for xy, the answer is 21, not 7 or 3.

ACT may show a system in a real scenario. Let t be adult tickets and s be student tickets. If 30 total tickets cost $246, with adult tickets $10 and student tickets $6, the system is t + s = 30 and 10t + 6s = 246. Multiply the first equation by 6 to get 6t + 6s = 180. Subtract from the cost equation: 4t = 66, so t = 16.5. That impossible half ticket tells you a number was copied wrong or a condition was misread; reasonableness checks matter.

Use that same setup with a total cost of $244: 10t + 6s = 244, subtract 180, and 4t = 64, so t = 16 and s = 14. The method is clean because the variables and units were defined before solving.

Systems can also describe lines. If two equations have the same slope but different intercepts, the lines are parallel and no ordered pair satisfies both. If one equation is just a multiple of the other, every point on that line works. ACT may ask how many solutions a system has, so do not automatically solve for x and y when the structure already answers the question.

ACT Modeling Traps

• If choices are numeric, backsolving can work, but solving directly is usually faster for a clean linear equation.
• If choices are expressions, transform your equation until it matches an equivalent form.
• If the prompt asks for "how many more," subtract after finding both quantities.
• If the prompt asks for a value of an expression such as 2x + y, compute that expression instead of reporting x.
• If your answer creates a negative count, fractional person, or impossible rate, reread the setup.

When time is tight, use answer choices deliberately. For a system asking for one variable, a numeric choice can be substituted into the cleaner equation first, then checked in the other equation. For a formula-choice question, simplify your expression toward the choices rather than expanding everything by habit.

The final check is substitution. Put your solution back into the original equation or system, not your simplified middle line. Substitution catches sign errors, lost constants, and answers that solved a different question from the one ACT asked.