Problem 115 - Introductory and Intermediate Algebra for College Students 4th

Question

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

Step-by-Step Solution

Answer

Solving linear inequalities and linear equations share similar steps of simplifying both sides of the equation or inequality and isolating the variable. The main difference appears when multiplication or division by a negative number occurs, as only inequalities need to reverse the inequality sign while equalities remain unchanged.

Step 1: Equations and Inequalities Similarities

Both linear equations and linear inequalities involve the same initial steps: simplifying both sides of the equation or inequality and then isolating the variable on one side. The operations of addition, subtraction, multiplication, and division are used to isolate the variable. Likewise, any transformation that maintains the equality of an equation or the inequality of an inequality can be applied.

Step 2: Understanding Differences

The primary difference between the two is how they handle multiplication or division by negative numbers. If you multiply or divide an inequality by a negative number, the sign of the inequality must be reversed. With linear equations, this step is not needed as multiplying or dividing by negative numbers do not change the equal sign.

Step 3: Example

Let's consider a linear equation, '2x = 6' and a linear inequality, '2x > 6'. For the equation, divide both sides by 2 to isolate x, the solution would be 'x = 3'. For the inequality, the process of isolating x is same i.e., we divide both sides by 2 but the inequality sign remains the same. The solution, in this case, would be 'x > 3'.