Solve each inequality. Give the solution set using interval notation.
$7x-2(x-3)\le 5(2-x)$

Verified step by step guidance

1. Start by expanding the expressions on both sides of the inequality: expand \$7x - 2(x - 3)\( and \)5(2 - x)$.
2. Distribute the multiplication over addition/subtraction inside the parentheses: \(7x - 2x + 6 \leq 10 - 5x\).
3. Combine like terms on the left side: \((7x - 2x)\) becomes \$5x$, so the inequality is \(5x + 6 \leq 10 - 5x\).
4. Add \$5x\( to both sides to get all \)x$ terms on one side: \(5x + 5x + 6 \leq 10\), which simplifies to \(10x + 6 \leq 10\).
5. Subtract 6 from both sides to isolate the term with \(x\): \(10x \leq 4\), then divide both sides by 10 to solve for \(x\).

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 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 across terms inside parentheses, such as a(b + c) = ab + ac. This is essential for simplifying expressions on both sides of the inequality before solving, ensuring all terms are combined correctly.

Interval Notation

Interval notation is a concise way to represent solution sets of inequalities using intervals. It uses parentheses for values not included (open intervals) and brackets for values included (closed intervals), clearly showing the range of solutions on the number line.