Solving linear inequalities

Solving Linear Inequalities

Linear inequalities are mathematical expressions that involve a linear function and an inequality sign. The inequality sign can be one of the following: < (less than), > (greater than), <= (less than or equal to), or >= (greater than or equal to). Solving linear inequalities is similar to solving linear equations, but with some important differences. The solution to a linear inequality is a range of values rather than a single value.

Understanding Linear Inequalities

A linear inequality can be written in the form:

ax + b < c, ax + b > c, ax + b <= c, or ax + b >= c

where a, b, and c are real numbers, and x is the variable we want to solve for. The solution set of a linear inequality includes all values of x that make the inequality true.

Steps to Solve Linear Inequalities

To solve a linear inequality, follow these steps:

1. Isolate the variable on one side of the inequality.
2. Simplify both sides if needed.
3. Determine the direction of the inequality.
4. Write the solution in interval notation or set notation.

Example 1: Solving a Simple Linear Inequality

Consider the inequality:

2x + 3 > 7

Step 1: Isolate the variable x.

Subtract 3 from both sides:

2x + 3 - 3 > 7 - 3

This simplifies to:

2x > 4

Step 2: Divide both sides by 2:

x > 2

The solution is x > 2. This means that any number greater than 2 will satisfy the inequality.

Example 2: Solving with a Negative Coefficient

Now consider the inequality:

-3x + 4 < 10

Step 1: Isolate x.

Subtract 4 from both sides:

-3x + 4 - 4 < 10 - 4

This simplifies to:

-3x < 6

Step 2: Divide both sides by -3. Remember to reverse the inequality sign:

x > -2

The solution is x > -2.

Example 3: Solving an Inequality with Variables on Both Sides

Consider the inequality:

5x - 1 > 2x + 8

Step 1: Move all terms involving x to one side and constant terms to the other side.

Subtract 2x from both sides:

5x - 2x - 1 > 8

This simplifies to:

3x - 1 > 8

Step 2: Add 1 to both sides:

3x - 1 + 1 > 8 + 1

This simplifies to:

3x > 9

Step 3: Divide both sides by 3:

x > 3

The solution is x > 3.

Example 4: Solving a Compound Inequality

Sometimes, you may encounter compound inequalities, which involve two inequalities joined by the word