In mathematics, an equation is a statement that asserts the equality of two expressions. These expressions are connected by the equal sign (=). The goal of solving an equation is to find the value(s) of the variable(s) that make the equation true.
Example: Solving a Simple Linear Equation
Consider the equation 2x + 3 = 11.
Step 1: Subtract 3 from both sides: 2x = 8.
Step 2: Divide both sides by 2: x = 4.
Answer: The solution to the equation is x = 4.
Inequalities are similar to equations, but instead of an equal sign, they use inequality signs (<, >, ≤, ≥). Inequalities express a range of possible solutions rather than a single solution. Solving an inequality involves finding all values of the variable that satisfy the inequality.
Example: Solving an Inequality
Consider the inequality 3x - 4 > 5.
Step 1: Add 4 to both sides: 3x > 9.
Step 2: Divide both sides by 3: x > 3.
Answer: The solution to the inequality is x > 3, meaning any value greater than 3 will satisfy the inequality.
These are equations and inequalities where the variable appears to the first power (i.e., the variable is not squared, cubed, etc.). Linear equations are of the form ax + b = 0, while linear inequalities can take forms like ax + b > 0 or ax + b < 0.
Example: Linear Equation
Solve the linear equation 5x - 7 = 18.
Step 1: Add 7 to both sides: 5x = 25.
Step 2: Divide both sides by 5: x = 5.
Answer: The solution to the equation is x = 5.
Quadratic equations are polynomial equations of degree 2, typically in the form ax² + bx + c = 0. Quadratic inequalities involve expressions where the variable is squared and can take forms like ax² + bx + c > 0.
Example: Solving a Quadratic Equation
Consider the quadratic equation x² - 5x + 6 = 0.
Step 1: Factor the quadratic expression: (x - 2)(x - 3) = 0.
Step 2: Set each factor to zero: x - 2 = 0 or x - 3 = 0.
Answer: The solutions are x = 2 and x = 3.
Absolute value equations involve the absolute value function, which measures the distance of a number from zero on a number line. These equations take the form |ax + b| = c, where c is a non-negative number.
Example: Absolute Value Equation
Solve the equation |2x - 3| = 7.
Step 1: Set up two cases: 2x - 3 = 7 and 2x - 3 = -7.
Step 2: Solve each case: x = 5 or x = -2.
Answer: The solutions are x = 5 and x = -2.
Solving basic equations involves a series of steps that can vary depending on the type of equation. However, some general principles apply to all types of equations:
Solving inequalities requires similar steps to solving equations, with some additional considerations:
Equations and inequalities are not just theoretical concepts; they have practical applications in various fields:
