📖 2. Linear Equations And Inequalities

📖 From Digital SAT Algebra • 657 questions available

About 2. Linear Equations And Inequalities

Definition:
A linear equation is an algebraic statement where the highest power of the variable is one, forming a straight line when graphed. Solving it means finding the specific value of the variable that makes the equation true, by performing inverse operations to isolate the variable on one side of the equals sign.

Working:
To solve, we use the addition and multiplication properties of equality to systematically undo operations around the variable. For example, we add the same number to both sides to cancel subtraction, or divide both sides by the same non-zero number to remove a coefficient.

Example:
Solve $3x + 5 = 20$.
Subtract 5 from both sides: $3x = 15$.
Divide both sides by 3: $x = 5$.
Thus, the solution is $x = 5$.

Reason:
This method ensures the equality remains balanced, as whatever operation we do to one side must be done to the other to maintain the 'balance' of the equation. It is the fundamental technique for finding unknown quantities in real-world problems.

Definition:
A linear inequality is a mathematical statement that compares two linear expressions using inequality symbols such as $<, >, \leq, \text{or} \geq$. Solving it means finding the range of possible values for the variable that make the inequality true, instead of a single fixed number.

Working:
We solve inequalities similarly to equations by isolating the variable, but with one key exception: if we multiply or divide by a negative number, we must reverse the inequality sign to keep the statement correct. The solution is often expressed as an interval or on a number line.

Example:
Solve $-2x + 4 \leq 10$.
Subtract 4 from both sides: $-2x \leq 6$.
Divide both sides by -2 and reverse the sign: $x \geq -3$.
Thus, the solution is all real numbers $x \geq -3$.

Reason:
Reversing the inequality when multiplying or dividing by a negative is necessary because it preserves the true order of numbers on the number line, ensuring the solution set correctly represents all values that satisfy the original condition.

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🔄 Last updated: 2026-08-27