linear inequality

Linear Inequality

A linear inequality is a mathematical expression that relates two expressions using an inequality symbol. Inequalities show how numbers compare to one another. They are like linear equations but with inequality signs instead of an equal sign.

Symbols of Inequality

There are four main inequality symbols:

• <: less than
• >: greater than
• ≤: less than or equal to
• ≥: greater than or equal to

Understanding Linear Inequalities

Linear inequalities involve variables like x or y. They can be written in the form: \(ax + b < c\), \(ax + b > c\), \(ax + b \le c\), or \(ax + b \ge c\). Here, a, b, and c are numbers. Let's go over some examples:

Example 1

Solve the inequality \(2x + 3 < 7\).

1. First, we subtract 3 from both sides:
   \(2x + 3 - 3 < 7 - 3\)
   Simplifies to \(2x < 4\).
2. Next, we divide by 2 on both sides:
   \(\frac{2x}{2} < \frac{4}{2}\)
   Simplifies to \(x < 2\).

Example 2

Solve the inequality \(4x - 5 > 3\).

1. Add 5 to both sides:
   \(4x - 5 + 5 > 3 + 5\)
   Simplifies to \(4x > 8\).
2. Divide by 4 on both sides:
   \(\frac{4x}{4} > \frac{8}{4}\)
   Simplifies to \(x > 2\).

Example 3

Solve the inequality \(-3x + 2 \le 11\).

1. Subtract 2 from both sides:
   \(-3x + 2 - 2 \le 11 - 2\)
   Simplifies to \(-3x \le 9\).
2. Divide by -3 on both sides and reverse the inequality sign:
   \(\frac{-3x}{-3} \ge \frac{9}{-3}\)
   Simplifies to \(x \ge -3\).

Graphing Linear Inequalities

We can show linear inequalities on a number line:

Summary of Key Points

• Linear inequalities use inequality symbols like <, >, ≤, and ≥.
• They can be solved similar to linear equations by performing arithmetic operations on both sides.
• When multiplying or dividing by a negative number, reverse the inequality sign.
• Graphing inequalities on a number line helps visualize the solution.