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

Verified step by step guidance

1. Start by writing down the inequality: $\frac{2x-3}{4}\ge \frac{3}{4}+\frac{1}{2}$.
2. To eliminate the denominators, multiply every term on both sides of the inequality by 4, the least common denominator, to simplify the expression.
3. After multiplying, simplify both sides by distributing and combining like terms where possible.
4. Isolate the variable $x$ on one side by subtracting or adding terms accordingly, and then solve for $x$ by dividing or multiplying as needed.
5. Once you have the inequality in the form $x\ge \mathrm{a}$ or $x\le \mathrm{a}$, graph the solution on a number line by shading the region that satisfies the inequality and using a closed or open circle depending on whether the inequality includes equality.

Key Concepts

Solving Linear Inequalities

Solving linear inequalities involves isolating the variable on one side to find the range of values that satisfy the inequality. Similar to equations, operations like addition, subtraction, multiplication, and division are used, but multiplying or dividing by a negative number reverses the inequality sign.

Combining Like Terms and Simplifying Expressions

Combining like terms means adding or subtracting terms with the same variable and exponent to simplify expressions. Simplifying helps to reduce the inequality to a more manageable form, making it easier to isolate the variable and solve.

Graphing Solution Sets on a Number Line

Graphing solution sets involves representing all values that satisfy the inequality on a number line. Use a solid dot for inclusive inequalities (≥ or ≤) and an open dot for strict inequalities (> or <), shading the region that represents all possible solutions.