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 represent the solution set in i

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.
$$8 x-11 \leq 3 x-13$$

Key Concepts

Interval Notation

Interval notation is a succinct way to represent subsets of the real numbers. It uses brackets [ ] to indicate that an endpoint is included (closed interval) and parentheses ( ) to indicate that it is not included (open interval). This notation is particularly useful for expressing the continuum of values that satisfy an inequality.

Linear Inequality

A linear inequality is similar to a linear equation but uses inequality symbols (such as <, ?, >, or ?) instead of an equal sign. It represents a range of possible solutions rather than a single value. Solving a linear inequality involves isolating the variable while considering the implications of operations such as multiplying or dividing by negatives, which can flip the inequality sign.

Solving Inequalities

The process involves applying algebraic principles such as addition, subtraction, multiplication, and division to both sides of the inequality to isolate the variable. A key point is that if you multiply or divide by a negative number, the direction of the inequality reverses. This method yields the correct range of values that satisfy the original inequality.

Graphing on a Number Line

Graphing the solution to an inequality on a number line provides a visual representation of the set of solutions. It involves marking the endpoints appropriately (with a closed circle for included endpoints and an open circle for excluded endpoints) and shading or drawing an arrow to indicate all the values that satisfy the inequality.

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