A linear equation is one in which the highest power of the variable is 1; its graph is a straight line, and its general form is \(ax + b = 0\). The goal when solving such an equation is to isolate the variable using inverse operations. For example, in \(3x + 7 = 22\), you subtract \(7\) from both sides to get \(3x = 15\), and then divide both sides by \(3\) to find \(x = 5\). When the equation contains parentheses, the distributive property is used first; in \(5(x - 2) = 3x + 4\), distributing gives \(5x - 10 = 3x + 4\), after which subtracting \(3x\) and adding \(10\) leads to \(2x = 14\) and finally \(x = 7\).
When fractions appear in a linear equation, multiplying every term by the least common denominator clears them so that standard solving methods can be applied. For instance, in \(\frac{x}{3} + 2 = 5\), you can subtract \(2\) from both sides to obtain \(\frac{x}{3} = 3\), and then multiply both sides by \(3\) to get \(x = 9\). Not every linear equation has exactly one solution, however. An equation such as \(2(x + 3) = 2x + 6\) simplifies to \(0 = 0\), a statement that is always true regardless of the value of \(x\); such an equation is said to have infinitely many solutions. By contrast, simplifying some equations leads to a contradiction like \(0 = 5\), which is never true and indicates that the equation has no solution.
Inequalities extend the idea of equations by comparing quantities with symbols such as \(<\), \(>\), \(\leq\), or \(\geq\). The symbols \(<\) and \(>\) represent strict inequalities, in which the boundary value is not included in the solution set, while \(\leq\) and \(\geq\) are non-strict and do include the boundary. Most inequality-solving steps mirror those used for equations, with one crucial exception: multiplying or dividing both sides by a negative number reverses the direction of the inequality sign. So if \(-2x > 6\), dividing by \(-2\) yields \(x < -3\). A compound inequality like \(-2 < 3x + 1 \leq 10\) can be solved by performing the same operation on all three parts at once, leading to \(-1 < x \leq 3\).
Solution sets are commonly written in interval notation, where parentheses indicate an excluded endpoint and brackets indicate an included endpoint. For example, the solution to \(3x - 4 \leq 11\) is \(x \leq 5\), which in interval notation is \((-\infty, 5]\). The solution to \(2x + 5 > 3x - 1\) is \(x < 6\), written as \((-\infty, 6)\). Absolute value equations and inequalities are another important family of problems. Since the absolute value \(|a|\) represents the distance of \(a\) from zero, it is always non-negative. The equation \(|x| = a\) for \(a \geq 0\) splits into two cases, \(x = a\) or \(x = -a\), while \(|x| < a\) becomes \(-a < x < a\) and \(|x| > a\) becomes \(x > a\) or \(x < -a\). Applied to \(|2x - 3| = 7\), for instance, the two cases give \(x = 5\) and \(x = -2\).