In all exercises, other than ∅, use interval notation to express solution sets and graph each so

step 1 We need to solve the given linear inequality and then we need to write the solution set in inter

In all exercises, other than ∅, use interval notation to...

In all exercises, other than $\varnothing,$ use interval notation to express solution sets and graph each solution set on a number line. In Exercises $27-48,$ solve each linear inequality.
$$5[3(2-3 x)-2(5-x)]-6[5(x-2)-2(4 x-3)]$$
$$<3 x+19$$

Key Concepts

Linear Inequalities

Linear inequalities involve expressions of degree one and use inequality symbols (such as <, >, ?, or ?) instead of an equals sign. The goal is to find all values of the variable that make the inequality true while carefully handling operations that may reverse the inequality direction when multiplying or dividing by negative numbers.

Distributive Property

The distributive property allows one to multiply a single term across terms inside parentheses. This property is critical to expand expressions and simplify equations or inequalities, transforming more complex expressions into simpler ones that are easier to solve.

Combining Like Terms

Combining like terms is the process of simplifying an expression by adding or subtracting coefficients of the same variable. This step is essential in reducing complex algebraic expressions to simpler forms when solving equations and inequalities.

Solving Inequalities

Solving inequalities involves isolating the variable on one side through operations such as addition, subtraction, multiplication, and division. Special attention must be paid to the direction of the inequality sign, especially when multiplying or dividing by a negative number.

Interval Notation

Interval notation is a method of representing the solution set of an inequality using intervals on the number line. It clearly denotes which endpoints are included or excluded, providing a concise way to describe continuous sets of numbers.

Graphing on a Number Line

Graphing on a number line involves marking the regions that represent the solution set of an inequality. This visual representation helps in understanding the range of values that satisfy the inequality and clarifies the inclusion or exclusion of boundary points.

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