Problem 135 - College Algebra Essentials

Question

Describe ways in which solving a linear inequality is similar to solving a linear equation.

Step-by-Step Solution

Answer

Solving a linear inequality is similar to solving a linear equation in many ways: both involve isolating the variable and we can use operations like addition, subtraction, multiplication, or division on both sides. The difference when solving a linear inequality is remembering to flip the inequality sign when multiplied or divided by a negative number.

Step 1: Understanding the Concepts

A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. A linear inequality, on the other hand, is simply a linear equation with an inequality sign instead of an equals sign.

Step 2: Solving a Linear Equation

The process of solving a linear equation usually involves simplifying the given equation by using properties of equality until the variable is isolated on one side. This process typically includes operations like addition, subtraction, multiplication, or division.

Step 3: Solving a Linear Inequality

To solve a linear inequality, the same steps are followed as with solving a linear equation, with one exception: when multiplying or dividing both sides of an inequality by a negative number, the direction of the inequality sign must be reversed. For instance, if we have \(-x > 1\), it becomes \(x < -1\) after multiplying by -1.

Step 4: Comparing Both Methods

Solving a linear inequality is very much like solving a linear equation. In both cases, the main goal is to isolate the variable. And in both cases, we can add, subtract, multiply, or divide both sides by the same value (except for division by zero). The main difference is that, when multiplied or divided by a negative number in an inequality, the inequality direction changes.