Solve each inequality. Give the solution set using interval notation.
$11 x \geq 2 (x - 4)$

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Start by writing down the inequality: $11x \geq 2(x - 4)$.

Distribute the 2 on the right side: $11x \geq 2x - 8$.

Subtract \$2x$ from both sides to get all $x$ terms on one side: $11x - 2x \geq -8$.

Simplify the left side: $9x \geq -8$.

Divide both sides by 9 (a positive number, so the inequality direction stays the same): $x \geq \frac{-8}{9}$, then express the solution set in interval notation as $\left[ \frac{-8}{9}, \infty \right)$.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 special care is needed when multiplying or dividing by negative numbers, as this reverses the inequality sign.

Distributive Property

The distributive property allows you to multiply a single term by each term inside parentheses. For example, 2(x - 4) becomes 2x - 8. This step is essential to simplify expressions and combine like terms when solving inequalities.

Interval Notation

Interval notation is a concise way to represent the solution set of inequalities. It uses parentheses and brackets to indicate open or closed intervals, showing all values that satisfy the inequality. For example, (3, ∞) means all numbers greater than 3.

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