A linear inequality is an inequality in which the variable appears only to the first power (i.e., it is not raised to any exponents greater than 1) and is not part of any higher-order operations like multiplication or division involving other variables. It resembles a linear equation but instead of equality, it uses one of the inequality signs (>, <, ≥, or ≤).

General Form of a Linear Inequality:

A linear inequality typically takes the following forms:

1. $ax + b > 0$
2. $ax + b \geq 0$
3. $ax + b < 0$
4. $ax + b \leq 0$

Where:

• $a$ and $b$ are constants.
• $x$ is the variable.
• The inequality sign can be one of >, <, ≥, or ≤.

Solving Linear Inequalities:

To solve linear inequalities, the goal is to find the range of values for $x$ that make the inequality true. This is similar to solving linear equations, but with a few important rules:

1. Addition/Subtraction: You can add or subtract the same number to both sides of the inequality without changing the inequality's direction.
2. Multiplication/Division: When multiplying or dividing by a positive number, the inequality sign stays the same. However, if you multiply or divide by a negative number, you must reverse the inequality sign.

Example:

Solve the inequality:
$3x - 5 < 7$

1. Add 5 to both sides:
   $3x < 12$

2. Divide by 3:
   $x < 4$

Thus, the solution is $x < 4$.

Graphing Linear Inequalities:

When graphing linear inequalities, the solution is often represented as a shaded region on a number line (for 1-variable) or on a coordinate plane (for 2-variable). The boundary line is drawn as either a solid line (for ≥ or ≤) or a dashed line (for > or <), and the region where the inequality holds true is shaded.

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Would you like more details or an example for graphing linear inequalities?

Here are some related questions for further learning:

1. How do we graph a two-variable linear inequality?
2. What is the difference between linear equations and linear inequalities?
3. How do we solve a system of linear inequalities?
4. How does multiplying by a negative number affect the inequality?
5. What are the practical applications of linear inequalities in real life?

Tip: Always be cautious when multiplying or dividing by negative numbers in inequalities, as it changes the direction of the inequality sign.