Solve each inequality. Give the solution set in interval notation. 8x-3x+2<2(x+7)

Verified step by step guidance

1. First, simplify both sides of the inequality. On the left side, combine like terms: \$8x - 3x + 2\( becomes \)5x + 2$.
2. Rewrite the inequality with the simplified left side: \$5x + 2 < 2(x + 7)$.
3. Next, distribute the 2 on the right side: \$2(x + 7)\( becomes \)2x + 14$.
4. Now, the inequality is \$5x + 2 < 2x + 14\(. To isolate the variable terms on one side, subtract \)2x\( from both sides: \)5x - 2x + 2 < 14\( which simplifies to \)3x + 2 < 14$.
5. Then, subtract 2 from both sides to isolate the term with \(x\): \$3x < 14 - 2\(, which simplifies to \)3x < 12\(. Finally, divide both sides by 3 to solve for \)x$: \(x < \frac{12}{3}\).

Key Concepts

Solving Linear Inequalities

Solving linear inequalities involves isolating the variable on one side to determine 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

Combining like terms means simplifying expressions by adding or subtracting terms with the same variable and exponent. This step reduces complexity and helps isolate the variable, making it easier to solve inequalities or equations.

Interval Notation

Interval notation is a concise way to represent solution sets of inequalities using parentheses and brackets. Parentheses indicate values not included (open interval), while brackets indicate values included (closed interval), clearly showing the range of solutions.