Solve each inequality. Give the solution set using interval notation. See Examples 8 and 9. 4x + 7———— ≤ 2x + 5 -3

Verified step by step guidance

1. Step 1: Start by isolating the variable term on one side of the inequality. Multiply both sides of the inequality by -3 to eliminate the fraction. Remember that multiplying or dividing by a negative number reverses the inequality sign.
2. Step 2: Distribute the -3 on the right side of the inequality to both terms inside the parentheses.
3. Step 3: Combine like terms by moving all terms involving x to one side of the inequality and constant terms to the other side.
4. Step 4: Simplify the inequality by combining like terms and isolating the variable x.
5. Step 5: Express the solution set in interval notation, considering the direction of the inequality.

Key Concepts

Inequalities

Inequalities are mathematical statements that express the relationship between two expressions that are not necessarily equal. They can be represented using symbols such as <, >, ≤, and ≥. Solving inequalities involves finding the values of the variable that make the inequality true, which often requires similar techniques to solving equations, but with additional considerations for the direction of the inequality when multiplying or dividing by negative numbers.

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 in the set. For example, (a, b) means all numbers between a and b, excluding a and b, while [a, b] includes both endpoints. This notation is particularly useful for expressing the solution sets of inequalities succinctly.

Combining Like Terms

Combining like terms is a fundamental algebraic technique used to simplify expressions by merging terms that have the same variable raised to the same power. This process is essential when solving inequalities, as it helps to isolate the variable on one side of the inequality. For instance, in the given inequality, combining terms on both sides will facilitate finding the solution set more efficiently.