Problem 26 - Algebra for College Students

Question

Express each interval as an inequalit using the variable \(x\). For example, we can express the inter val \([5, \infty)\) as \(x \geq 5\). \(5 x-3<-3\)

Step-by-Step Solution

Answer

The inequality is \(x < 0\).

Step 1: Simplify the inequality

Start by adding 3 to both sides of the inequality:\[5x - 3 + 3 < -3 + 3\]This simplifies to:\[5x < 0\]

Step 2: Solve for x

Now divide both sides of the inequality by 5 to isolate the variable x:\[x < \frac{0}{5}\]This reduces to:\[x < 0\]

Key Concepts

Solving inequalities

In algebra, an inequality is a mathematical statement that compares two expressions. Understanding how to solve inequalities is crucial because it allows us to determine the values that satisfy a given condition. An inequality uses symbols such as <, >, ≤, or ≥ to show the relationship between two expressions. The key steps in solving inequalities mirror those used in solving equations, but with special consideration of certain rules when multiplying or dividing by negative numbers.
For example, after simplifying an inequality, such as \[5x - 3 < -3\]we first isolate the variable by performing arithmetic operations to both sides.

• Always perform the same operation on both sides to maintain balance.
• If you multiply or divide both sides by a negative number, remember to flip the inequality sign.

By following these steps, you can find a range of values for the variable that satisfies the inequality.

Interval notation

Interval notation is a way of writing subsets of the real number line. This notation is particularly useful when representing the solution to an inequality in a concise form.
The simplest example is converting an inequality into interval notation. For instance, for an inequality like\[x < 0\] the interval notation is\[(-\infty, 0)\].

• The parentheses \((\ )\) indicate that the endpoints are not included in the interval (open interval).
• Brackets \([\ ]\) would be used if the endpoints were part of the solution (closed interval).

Interval notation provides an easy-to-read format that conveys exactly where your solution set lies on the real number line, making it a preferred method for expressing solutions to inequalities.

Variable isolation

Variable isolation is a fundamental concept in algebra that involves manipulating an equation or inequality to get the variable by itself on one side of the expression. This process allows us to clearly see what values the variable can take.
To isolate a variable in an inequality such as \[5x < 0\],we can perform operations that allow us to simplify the expression:

• Addition or subtraction to remove constants from the side with the variable.
• Division or multiplication to manage coefficients attached to the variable.

In the example given, dividing both sides by 5 isolates the variable, leading to \[x < 0\].This step-by-step approach to simplification helps you systematically find solutions to inequalities while ensuring the mathematical statement remains true.