Solve and graph the solution set on a number line: (2x−3)/4 ≥ 3x/4 + 1/2

Verified step by step guidance

1. Multiply every term by 4 to eliminate the fractions: \(4 \times \frac{2x - 3}{4} \geq 4 \times \frac{3x}{4} + 4 \times \frac{1}{2}\).
2. Simplify the equation: \(2x - 3 \geq 3x + 2\).
3. Rearrange the inequality to isolate \(x\): Subtract \(3x\) from both sides to get \(2x - 3x - 3 \geq 2\).
4. Simplify further: \(-x - 3 \geq 2\).
5. Add 3 to both sides to isolate \(-x\): \(-x \geq 5\), then multiply by -1 and reverse the inequality sign to get \(x \leq -5\).

Key Concepts

Inequalities

Inequalities are mathematical expressions that show the relationship between two values when they are not equal. They use symbols such as '≥' (greater than or equal to) and '≤' (less than or equal to) to indicate the direction of the relationship. Understanding how to manipulate inequalities is crucial for solving them, as the rules can differ from those of equations, especially when multiplying or dividing by negative numbers.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions to isolate variables or solve equations and inequalities. This includes operations such as adding, subtracting, multiplying, and dividing both sides of an inequality by the same number, as well as combining like terms. Mastery of these techniques is essential for effectively solving the given inequality and finding the solution set.

Graphing on a Number Line

Graphing on a number line is a visual representation of the solution set of an inequality. It involves marking points or intervals that satisfy the inequality, using open or closed circles to indicate whether endpoints are included or excluded. Understanding how to accurately represent solutions on a number line helps in visualizing the range of values that meet the conditions of the inequality.