One-variable equations and inequalities Knowledge-Based Study

Study notes

1. Foundations of Linear Equations

   A linear equation in one variable is an equality that can be written in the standard form $ax + b = 0$, where a and b are real numbers and $a \neq 0$. The value of the variable that makes the equation true is called the solution or root.

   The Addition and Multiplication Properties

   Solving an equation involves finding equivalent equations by isolating the variable:

   • Addition and Subtraction Property: If $a = b$, then $a + c = b + c$ and $a - c = b - c$.
   • Multiplication and Division Property: If $a = b$ and $c \neq 0$, then $a \cdot c = b \cdot c$ and $\frac{a}{c} = \frac{b}{c}$.

   Step-by-Step Solving Process

   1. Clear fractions or decimals by multiplying through by the least common denominator.
   2. Simplify both sides by applying the distributive property and combining like terms.
   3. Use addition or subtraction to collect all variable terms on one side and constant terms on the other.
   4. Isolate the variable using multiplication or division.

   Types of Linear Equations

   When solving linear equations, simplifying terms can lead to three distinct outcomes:

   • Conditional Equation: The equation has exactly one solution. For example, $3x - 5 = 7$ simplifies to $3x = 12$, giving $x = 4$.
   • Identity: The equation is true for all real numbers within the domain. Simplifying both sides yields an unconditionally true statement, such as $0 = 0$ or $5 = 5$. In this case, the solution set is all real numbers, denoted as $\mathbb{R}$.
   • Contradiction: The equation has no solution. Simplifying both sides leads to a false statement, such as $0 = 7$. In this case, no value of the variable satisfies the equality, and the solution set is the empty set, denoted as $\emptyset$.

   Recognizing these outcomes prevents calculation errors when variables cancel out entirely during algebraic manipulation.

   Properties of Inequalities

   A linear inequality in one variable relates algebraic expressions using inequality symbols: $<$, $\le$, $>$, or $\ge$.

   Most properties of equations carry over directly to inequalities, with one essential exception:

   • Adding or subtracting the same number on both sides preserves the inequality direction: if $a < b$, then $a + c < b + c$.
   • Multiplying or dividing both sides by a positive number preserves the inequality direction: if $a < b$ and $c > 0$, then $ac < bc$.
   • Critical Rule: Multiplying or dividing both sides by a negative number reverses the inequality direction: if $a < b$ and $c < 0$, then $ac > bc$.

   Expressing Solution Sets

   Inequality solutions are generally intervals rather than single points. For example, the inequality $x \ge 3$ is represented in interval notation as $[3, \infty)$, where the square bracket indicates that 3 is included in the solution set.

   Worked Example and Analysis

   Consider the inequality:

   $2(3 - x) \le 4x + 18$

   Step 1: Distribute

   Apply the distributive property on the left side:

   $6 - 2x \le 4x + 18$

   Step 2: Collect Variable Terms

   Subtract $4x$ from both sides:

   $6 - 6x \le 18$

   Step 3: Collect Constant Terms

   Subtract 6 from both sides:

   $-6x \le 12$

   Step 4: Divide by a Negative Coefficient

   Divide both sides by $-6$ and reverse the inequality sign:

   $x \ge -2$

   Conclusion

   The solution set in interval notation is $[-2, \infty)$. Any value greater than or equal to $-2$ satisfies the original statement.