Solve each inequality. Give the solution set in interval notation. See Examples 1 and 2. 8x-3x+2<2(x+7)

Verified step by step guidance

1. Combine like terms on the left side of the inequality: $8x-3x+2<2(x+7)$ becomes $5x+2<2(x+7)$.
2. Distribute the 2 on the right side of the inequality: $2(x+7)$ becomes $2x+14$.
3. Rewrite the inequality with the distributed terms: $5x+2<2x+14$.
4. Subtract $2x$ from both sides to isolate terms with $x$ on one side: $5x-2x+2<14$ becomes $3x+2<14$.
5. Subtract 2 from both sides to further isolate the term with $x$: $3x<12$.

Key Concepts

Inequalities

Inequalities are mathematical statements that express the relationship between two expressions that are not necessarily equal. They use symbols such as <, >, ≤, and ≥ to indicate whether one side is less than, greater than, or equal to the other. Solving inequalities involves finding the values of the variable that make the inequality true.

Interval Notation

Interval notation is a way of representing a set of numbers between two endpoints. It uses parentheses and brackets to indicate whether the endpoints are included (closed interval) or excluded (open interval). For example, (a, b) means all numbers between a and b, not including a and b, while [a, b] includes both endpoints.

Combining Like Terms

Combining like terms is a fundamental algebraic process that simplifies expressions by merging terms that have the same variable raised to the same power. This step is crucial when solving inequalities, as it helps to reduce the expression to a more manageable form, making it easier to isolate the variable and determine the solution set.