How to Solve an Inequality

Understanding Inequalities and their Types

In mathematics, an inequality is a statement that two values or expressions are not equal. Instead, one value or expression is greater than or less than the other. Inequalities are denoted by symbols such as “<" (less than), ">” (greater than), “≤” (less than or equal to), and “≥” (greater than or equal to).

There are several types of inequalities that you may encounter in mathematics. These include:

• Linear Inequalities: These are inequalities that involve linear expressions, which are expressions that involve only variables raised to the first power and constant terms. Examples of linear inequalities include 2x + 3 < 7 and 5 - y > 2.

• Quadratic Inequalities: These are inequalities that involve quadratic expressions, which are expressions that involve variables raised to the second power. Examples of quadratic inequalities include x^2 – 3x + 2 < 0 and 2y^2 + 5y ≥ 3.

• Rational Inequalities: These are inequalities that involve rational expressions, which are expressions that involve fractions. Examples of rational inequalities include (x + 1)/(x – 2) < 0 and (y^2 - 4)/(y - 2) > 0.

• Absolute Value Inequalities: These are inequalities that involve absolute value expressions, which are expressions that give the distance of a value from zero. Examples of absolute value inequalities include |x – 3| ≤ 2 and |y + 1| > 4.

Understanding the type of inequality you are working with is important because it determines the method you will use to solve the inequality. Linear inequalities, for example, can be solved using basic operations like addition and subtraction, while quadratic inequalities may require factoring or using the quadratic formula. Rational inequalities may require finding the critical points and testing intervals, and absolute value inequalities may require breaking the inequality into two separate cases.

Solving Simple Inequalities using Basic Operations

Simple inequalities are those that involve only one variable and basic operations such as addition, subtraction, multiplication, and division. To solve a simple inequality, follow these steps:

1. Isolate the variable term on one side of the inequality.

   Use basic operations to move all variable terms to one side of the inequality and all constant terms to the other side.

2. Flip the inequality sign, if necessary.

   If you multiply or divide both sides of the inequality by a negative number, you must flip the inequality sign.

3. Simplify the inequality.

   Combine like terms and simplify both sides of the inequality.

4. Write the solution in set-builder notation or interval notation.

   The solution to an inequality is the set of all values that make the inequality true. You can write the solution in set-builder notation or interval notation.

Here is an example of solving a simple inequality using basic operations:

Solve for x: 2x – 4 ≤ 6

1. Isolate the variable term on one side of the inequality.

   Add 4 to both sides: 2x ≤ 10

2. Flip the inequality sign, if necessary.

   No need to flip the inequality sign.

3. Simplify the inequality.

   Divide both sides by 2: x ≤ 5

4. Write the solution in set-builder notation or interval notation.

   The solution is all values of x that are less than or equal to 5: {x | x ≤ 5} or (-∞, 5].

Solving Multi-Step Inequalities using Inverse Operations

Multi-step inequalities are those that involve more than one operation to solve. To solve a multi-step inequality, follow these steps:

1. Simplify both sides of the inequality using the distributive property or combining like terms.

2. Isolate the variable on one side of the inequality by performing inverse operations.

   Perform inverse operations to move all variable terms to one side of the inequality and all constant terms to the other side. Remember to perform the same operation on both sides of the inequality.

3. Flip the inequality sign, if necessary.

   If you multiply or divide both sides of the inequality by a negative number, you must flip the inequality sign.

4. Simplify the inequality.

   Combine like terms and simplify both sides of the inequality.

5. Write the solution in set-builder notation or interval notation.

   The solution to an inequality is the set of all values that make the inequality true. You can write the solution in set-builder notation or interval notation.

Here is an example of solving a multi-step inequality using inverse operations:

Solve for x: 2x + 3 > 7x – 4

1. Simplify both sides of the inequality.

   2x + 3 > 7x – 4 becomes 6x < 7.

2. Isolate the variable on one side of the inequality by performing inverse operations.

   Subtract 3 from both sides: 2x > 7x – 7.

   Subtract 7x from both sides: -5x > -7.

   Divide both sides by -5: x < 7/5.

3. Flip the inequality sign, if necessary.

   No need to flip the inequality sign.

4. Simplify the inequality.

   No need to simplify further.

5. Write the solution in set-builder notation or interval notation.

   The solution is all values of x that are less than 7/5: {x | x < 7/5} or (-∞, 7/5).

Solving Inequalities with Absolute Values

Absolute value inequalities are those that involve absolute value expressions. To solve an absolute value inequality, follow these steps:

1. Isolate the absolute value expression on one side of the inequality.

   If the absolute value expression is on one side of the inequality, leave it as is. If the absolute value expression is on both sides of the inequality, isolate it on one side by subtracting the same quantity from both sides of the inequality.

2. Break the inequality into two cases.

   One case is when the expression inside the absolute value is positive, and the other case is when the expression inside the absolute value is negative.

3. Solve each case separately.

   For the case where the expression inside the absolute value is positive, simplify the absolute value expression by removing the absolute value bars. For the case where the expression inside the absolute value is negative, simplify the absolute value expression by multiplying by -1 and then removing the absolute value bars.

4. Write the solution in set-builder notation or interval notation.

   The solution to an inequality is the set of all values that make the inequality true. You can write the solution in set-builder notation or interval notation.

Here is an example of solving an absolute value inequality:

Solve for x: |2x – 5| ≤ 3

1. Isolate the absolute value expression on one side of the inequality.

   2x – 5 ≤ 3 and 2x – 5 ≥ -3.

2. Break the inequality into two cases.

   Case 1: 2x – 5 ≤ 3

   Case 2: 2x – 5 ≥ -3

3. Solve each case separately.

   Case 1: 2x – 5 ≤ 3

   Add 5 to both sides: 2x ≤ 8

   Divide both sides by 2: x ≤ 4

   Case 2: 2x – 5 ≥ -3

   Add 5 to both sides: 2x ≥ 2

   Divide both sides by 2: x ≥ 1

4. Write the solution in set-builder notation or interval notation.

   The solution is all values of x that are between 1 and 4, inclusive: {x | 1 ≤ x ≤ 4} or [1, 4].

Graphical Representation of Inequalities and their Solutions

Inequalities can also be represented graphically on a number line or in the coordinate plane. The solutions to an inequality are the values that make the inequality true, and these values are represented by shaded regions on the graph.

To graph an inequality on a number line:

1. Identify the variable in the inequality.

2. Determine the direction of the inequality.

   If the inequality is “greater than” or “greater than or equal to,” shade to the right of the point. If the inequality is “less than” or “less than or equal to,” shade to the left of the point.

3. Mark the point that represents the value of the variable in the inequality.

4. Draw a shaded region to represent the set of all values that satisfy the inequality.

Here is an example of graphing an inequality on a number line:

Graph the inequality x > 2.

1. Identify the variable in the inequality.

   The variable is x.

2. Determine the direction of the inequality.

   The inequality is “greater than,” so shade to the right of the point.

3. Mark the point that represents the value of the variable in the inequality.

   Mark the point 2 on the number line.

4. Draw a shaded region to represent the set of all values that satisfy the inequality.

   Shade to the right of the point 2 on the number line.

To graph an inequality in the coordinate plane:

1. Rewrite the inequality in slope-intercept form.

2. Graph the line that represents the equation of the inequality.

3. Determine which side of the line represents the set of all points that satisfy the inequality.

   If the inequality is “greater than” or “greater than or equal to,” shade above the line. If the inequality is “less than” or “less than or equal to,” shade below the line.

4. The shaded region represents the set of all points that satisfy the inequality.

Here is an example of graphing an inequality in the coordinate plane:

Graph the inequality y < 2x + 3.

1. Rewrite the inequality in slope-intercept form.

   y < 2x + 3 becomes y - 2x < 3.

2. Graph the line that represents the equation of the inequality.

   Graph the line y = 2x + 3.

3. Determine which side of the line represents the set of all points that satisfy the inequality.

   The inequality is “less than,” so shade below the line.

4. The shaded region represents the set of all points that satisfy the inequality.