5 Ways to Solve Linear Inequalities

Understanding Linear Inequalities

Linear inequalities are mathematical expressions that compare two linear expressions using inequality signs such as <, >, ≤, or ≥. Solving linear inequalities is a fundamental concept in algebra and is used to find the range of values that satisfy the given inequality. In this article, we will discuss five ways to solve linear inequalities.

Method 1: Addition and Subtraction

One of the simplest ways to solve linear inequalities is by adding or subtracting the same value to both sides of the inequality. This method is used to isolate the variable on one side of the inequality.

For example, consider the inequality:

2x - 3 > 5

To solve this inequality, we can add 3 to both sides:

2x - 3 + 3 > 5 + 3

This simplifies to:

2x > 8

Next, we can divide both sides by 2:

x > 4

Therefore, the solution to the inequality is x > 4.

💡 Note: When adding or subtracting the same value to both sides of an inequality, the direction of the inequality sign remains the same.

Method 2: Multiplication and Division

Another way to solve linear inequalities is by multiplying or dividing both sides of the inequality by the same value. However, when multiplying or dividing by a negative number, the direction of the inequality sign is reversed.

For example, consider the inequality:

x/2 < 3

To solve this inequality, we can multiply both sides by 2:

x < 6

However, if we multiply both sides by -2, the direction of the inequality sign is reversed:

-x > -6

Therefore, the solution to the inequality is x < 6 or x > 6.

💡 Note: When multiplying or dividing both sides of an inequality by a negative number, the direction of the inequality sign is reversed.

Method 3: Factoring

Factoring is a method used to solve linear inequalities that involve quadratic expressions. To factor a quadratic expression, we need to find two binomials whose product is equal to the quadratic expression.

For example, consider the inequality:

x^2 - 4x - 5 ≤ 0

To solve this inequality, we can factor the quadratic expression:

(x - 5)(x + 1) ≤ 0

This inequality is true when either (x - 5) ≤ 0 or (x + 1) ≤ 0.

Solving for the first factor, we get:

x - 5 ≤ 0 –> x ≤ 5

Solving for the second factor, we get:

x + 1 ≤ 0 –> x ≥ -1

Therefore, the solution to the inequality is x ≤ 5 or x ≥ -1.

Method 4: Graphing

Graphing is a visual method used to solve linear inequalities. To graph an inequality, we first graph the corresponding equation on a number line. Then, we use the inequality sign to determine which side of the line satisfies the inequality.

For example, consider the inequality:

x + 2 > 0

To graph this inequality, we first graph the equation x + 2 = 0 on a number line.

x + 2 = 0 –> x = -2

Since the inequality sign is >, we shade the region to the right of x = -2.

Therefore, the solution to the inequality is x > -2.

Method 5: Using a Number Line

A number line is a visual representation of the real number system. To solve a linear inequality using a number line, we first plot the critical points on the number line. Then, we use the inequality sign to determine which region satisfies the inequality.

For example, consider the inequality:

2x - 3 ≤ 5

To solve this inequality, we first plot the critical point x = 4 on the number line.

x = 4 –> 2x - 3 = 5

Since the inequality sign is ≤, we shade the region to the left of x = 4.

Therefore, the solution to the inequality is x ≤ 4.

Solving linear inequalities is an essential skill in algebra and is used to model real-world problems. By using these five methods, you can solve linear inequalities with confidence.

In summary, solving linear inequalities involves using various methods such as addition and subtraction, multiplication and division, factoring, graphing, and using a number line. By mastering these methods, you can solve linear inequalities and apply them to real-world problems.