Solve each inequality. Give the solution set in interval notation. See Examples 1 and 2. (4x+7)/-3≤2x+5

Verified step by step guidance

1. Start by isolating the variable term on one side. Multiply both sides of the inequality by -3 to eliminate the fraction, remembering to reverse the inequality sign: $(4x+7)\le -3(2x+5)$.
2. Distribute the -3 on the right side: $(4x+7)\le -6x-15$.
3. Combine like terms by adding $6x$ to both sides: $4x+6x+7\le -15$.
4. Simplify the left side: $10x+7\le -15$.
5. Subtract 7 from both sides to isolate the term with $x$: $10x\le -22$.

Key Concepts

Inequalities

Inequalities are mathematical statements that compare two expressions, indicating that one is less than, greater than, less than or equal to, or greater than or equal to the other. In this case, the inequality involves a rational expression and a linear expression, which requires understanding how to manipulate both sides to isolate the variable.

Interval Notation

Interval notation is a way of representing a set of numbers between two endpoints. It uses parentheses to indicate that an endpoint is not included in the set and brackets to indicate that it is included. Understanding how to express solution sets in interval notation is crucial for conveying the range of values that satisfy the inequality.

Solving Rational Inequalities

Solving rational inequalities involves finding the values of the variable that make the inequality true, often requiring the identification of critical points where the expression equals zero or is undefined. This process typically includes testing intervals between these points to determine where the inequality holds, which is essential for accurately determining the solution set.