In all exercises, other than exercises with no solution, use interval notation to express solution sets and graph each solution set on a number line. In Exercises 27–50, solve each linear inequality. 4(x + 1) + 2 ≥ 3x + 6

Verified step by step guidance

1. Start by expanding the left side of the inequality: distribute the 4 across the terms inside the parentheses. This gives you \(4 \cdot x + 4 \cdot 1\), so rewrite the inequality as \(4x + 4 + 2 \geq 3x + 6\).
2. Combine like terms on the left side: add 4 and 2 to simplify the expression to \(4x + 6 \geq 3x + 6\).
3. Next, isolate the variable terms on one side. Subtract \$3x$ from both sides to get \(4x - 3x + 6 \geq 6\), which simplifies to \(x + 6 \geq 6\).
4. Then, isolate \(x\) by subtracting 6 from both sides: \(x + 6 - 6 \geq 6 - 6\), which simplifies to \(x \geq 0\).
5. Express the solution in interval notation. Since \(x\) is greater than or equal to 0, the solution set is \([0, \infty)\), which you can graph on a number line by shading all values from 0 to positive infinity, including 0.

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, you perform operations like addition, subtraction, multiplication, or division, but must reverse the inequality sign when multiplying or dividing by a negative number.

Interval Notation

Interval notation is a concise way to represent solution sets of inequalities using parentheses and brackets. Parentheses indicate that an endpoint is not included, while brackets mean it is included. For example, [2, 5) represents all numbers from 2 to 5, including 2 but excluding 5.

Graphing Solution Sets on a Number Line

Graphing solution sets involves marking the range of values that satisfy the inequality on a number line. Use solid dots for included endpoints and open dots for excluded endpoints, shading the region that represents all solutions. This visual helps understand the solution's scope and boundaries.