Algebraic Concepts, Linear Equations & Inequalities

Algebraic Concepts, Linear Equations & Inequalities

Algebraic reasoning questions test your ability to translate real-world scenarios into mathematical models, isolate unknown variables, and manipulate expressions accurately. Mastering linear relations provides a solid foundation for both the HSPT Mathematics subtest and high school mathematics coursework, and algebra items reward students who work methodically on scratch paper under the no-calculator rule.

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Signed Numbers & Order of Operations

Signed numbers (positive and negative values) follow four core rules:

• Adding same signs: Add absolute values, keep the sign (-4 + (-7) = -11).
• Adding different signs: Subtract absolute values; take the larger number's sign (5 + (-9) = -4).
• Subtracting a negative: Convert to addition (3 - (-8) = 3 + 8 = 11).
• Multiplying/dividing: Same signs → positive; different signs → negative ((-6)(-4) = 24, -15 ÷ 3 = -5).

Simplify with PEMDAS (Parentheses, Exponents, Multiplication/Division, Addition/Subtraction); same-rank operations run left to right.

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Evaluating Algebraic Expressions & Exponents

To evaluate an algebraic expression, substitute given numerical values for each variable and simplify using PEMDAS.

Key Rule for Negative Base Exponents:

• (-x)² = (-x) · (-x) = +x² (Negative sign inside parentheses is squared).
• -x² = -(x · x) = -x² (Negative sign outside is applied after squaring).

Perfect Squares and Square Roots

Memorize the perfect squares through 15² (1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225) so roots become instant recall: √144 = 12.

Step-by-Step Worked Example 1

Problem: Evaluate 3x² - 2xy + y³ when x = -3 and y = 2.

1. Substitute values with parentheses: 3(-3)² - 2(-3)(2) + (2)³
2. Evaluate Exponents:
   • (-3)² = 9 → 3(9) = 27
   • (2)³ = 8
   • Expression: 27 - 2(-3)(2) + 8
3. Perform Multiplication:
   • 2(-3)(2) = -12
   • Expression: 27 - (-12) + 8
4. Simplify Addition/Subtraction:
   • 27 + 12 + 8 = 47

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Solving Multi-Step Linear Equations

To solve linear equations, isolate the target variable on one side of the equals sign using inverse operations:

1. Clear Parentheses: Apply the Distributive Property: a(b + c) = ab + ac.
2. Clear Fractions: Multiply every term on both sides by the Least Common Denominator (LCD).
3. Combine Like Terms: Group variable terms together and constant terms together on each respective side.
4. Isolate Variable: Use addition/subtraction to move constants to one side and variable terms to the other, then divide by the coefficient.

Combining Like Terms

Only like terms — terms with identical variables raised to identical exponents — can be combined: 5x + 3x = 8x, but 5x + 3x² must stay as written. Constant terms combine separately from variable terms, and each term keeps the sign directly in front of it when it moves.

Equation Solving Reference

Ratio and Proportion Setup

A proportion equates two ratios (a/b = c/d) and is solved by cross-multiplication: a · d = b · c. Keep matching units in matching positions.

• Example: 3 notebooks cost $7.50; cost of 10: 3/7.50 = 10/x → 3x = 75 → x = $25.

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Word Problem Translation Guide

Translating English sentences into algebraic equations requires converting operation keywords into mathematical symbols:

Word Order Warning: In subtraction, "7 less than x" translates to x - 7, NOT 7 - x. The phrase "less than" reverses the written order.

Step-by-Step Worked Example 2

Problem: Four times a number decreased by 9 is equal to twice the number increased by 15. Find the number.

1. Define Variable: Let n = the unknown number.
2. Translate Equation: 4n - 9 = 2n + 15
3. Solve for n:
   • Subtract 2n from both sides: 2n - 9 = 15
   • Add 9 to both sides: 2n = 24
   • Divide by 2: n = 12

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Solving and Graphing Linear Inequalities

Linear inequalities use the order symbols <, ≤, >, and ≥.

THE CRITICAL INEQUALITY RULE: Whenever you multiply or divide both sides of an inequality by a negative number, you MUST flip the direction of the inequality sign (< becomes >, ≤ becomes ≥, etc.).

Step-by-Step Worked Example 3

Problem: Solve and state the solution set for -5x + 12 ≥ 37.

1. Subtract 12 from both sides: -5x ≥ 25
2. Divide both sides by -5 and FLIP the inequality sign: x ≤ 25 / (-5) x ≤ -5
3. Graphing on a Number Line: Closed circle at -5 with a shaded arrow extending to the left.

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Common HSPT Traps

• Sign errors: Losing a negative when squaring (-3)², or forgetting that subtracting a negative becomes addition.
• Distributing negatives: -2(x - 5) = -2x + 10, not -2x - 10 — the negative factor hits every term inside the parentheses.
• Combining unlike terms: 3x² + 2x cannot be simplified; only terms with identical variables and exponents combine.
• Forgetting to flip the inequality: When dividing -3x > 15 by -3, the sign reverses to give x < -5, not x > -5.