Solve each equation or inequality. | 4x+ 3| > 0

Verified step by step guidance

1. Step 1: Understand the absolute value inequality |4x + 3| > 0. This inequality states that the expression inside the absolute value, 4x + 3, must be greater than 0.
2. Step 2: Recognize that an absolute value expression |A| is greater than 0 if A is not equal to 0. Therefore, we need to solve the inequality 4x + 3 ≠ 0.
3. Step 3: Solve the equation 4x + 3 = 0 to find the value of x that makes the expression inside the absolute value equal to zero.
4. Step 4: Subtract 3 from both sides of the equation to isolate the term with x: 4x = -3.
5. Step 5: Divide both sides by 4 to solve for x: x = -3/4. Therefore, the solution to the inequality |4x + 3| > 0 is all real numbers except x = -3/4.

Key Concepts

Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction. It is denoted by vertical bars, such as |x|. For example, |4| = 4 and |-4| = 4. Understanding absolute value is crucial for solving equations and inequalities that involve it, as it can lead to two possible cases based on the sign of the expression inside the absolute value.

Inequalities

Inequalities express a relationship between two expressions that are not necessarily equal. They use symbols such as '>', '<', '≥', and '≤' to indicate whether one side is greater than, less than, or equal to the other. In the context of the given problem, solving the inequality |4x + 3| > 0 requires determining the values of x for which the expression is not equal to zero, leading to a range of solutions.

Solution Sets

A solution set is the collection of all values that satisfy a given equation or inequality. For the inequality |4x + 3| > 0, the solution set includes all real numbers except those that make the expression inside the absolute value equal to zero. Understanding how to express and interpret solution sets is essential for effectively communicating the results of solving algebraic problems.