Two-Step Inequalities

This page focuses on solving two-step inequalities, which are more complex than one-step inequalities and require multiple operations to solve. It provides a step-by-step approach to solving these types of problems, which is crucial for students learning about solving inequalities by multiplication and division.

The page begins with two examples of two-step inequalities:

1. 4x + 1 ≤ 13
2. -3x + 2 > 11

For each example, the solving process is broken down into clear steps:

Example: Solving 4x + 1 ≤ 13 Step 1: Subtract 1 from both sides: 4x ≤ 12 Step 2: Divide both sides by 4: x ≤ 3

Example: Solving -3x + 2 > 11 Step 1: Subtract 2 from both sides: -3x > 9 Step 2: Divide both sides by -3 (and flip the inequality sign): x < -3

The document emphasizes the order of operations when solving two-step inequalities:

Highlight: Always perform addition or subtraction first, followed by multiplication or division.

It also reiterates the important rule about flipping the inequality sign when dividing by a negative number, which is a key concept in solving inequalities using multiplication and division.

The page concludes with graphing the solutions of these two-step inequalities on a number line, reinforcing the concepts of open and closed circles introduced on the previous page.

This comprehensive approach to solving and graphing two-step inequalities provides students with a solid foundation for understanding more complex inequality problems and prepares them for advanced topics in algebra.

Graphing and Solving Inequalities

This page introduces the fundamental concepts of inequalities, their symbols, and how to graph them on a number line. It covers essential information for students learning to solve inequalities with multiplication and division.

The page begins by presenting the four main inequality symbols:

• Greater than (>)
• Less than (<)
• Greater than or equal to (≥)
• Less than or equal to (≤)

Vocabulary: Inequality symbols are mathematical symbols used to show the relationship between two expressions that are not equal.

The document then explains how to graph inequalities on a number line, emphasizing the difference between open and closed circles:

Highlight: Open circles (○) are used for strict inequalities (> or <), while closed circles (●) represent inequalities that include equality (≥ or ≤).

Several examples of graphing inequalities are provided, such as x > 0 and x > -2, to illustrate the correct use of open and closed circles on a number line.

The page then delves into solving one-step inequalities, covering addition, subtraction, multiplication, and division. It emphasizes a crucial rule when solving inequalities:

Definition: When multiplying or dividing an inequality by a negative number, the inequality sign must be flipped.

This rule is particularly important for solving inequalities using multiplication and division. The document provides examples to demonstrate this concept, such as:

Example: -7x > 63 Dividing both sides by -7 (a negative number), we flip the sign: x < -9

The page concludes with a note on solving inequalities with addition and subtraction, stating that the inequality sign does not change in these operations unless adding a negative number, which is equivalent to subtraction.