Solving Inequalities

Inequalities are mathematical expressions that compare the values of two expressions, indicating whether one is greater than, less than, or equal to the other.

There are four types of inequalities:

  • Less than (<)
  • Greater than (>)
  • Less than or equal to (≤)
  • Greater than or equal to (≥)

For example, 3 < 5 means that 3 is less than 5, while 7 \geq 4 means that 7 is greater than or equal to 4.

Properties of Inequalities

Inequalities share some properties with equations. You can add or subtract the same value from both sides of an inequality without changing the inequality’s validity.

For example, if x > y, then x + 3 > y + 3. Likewise, you can multiply or divide both sides of an inequality by a positive number without affecting the inequality’s direction.

However, when multiplying or dividing by a negative number, the inequality’s direction reverses. For example, if x < y and both sides are multiplied by -2, then -2x > -2y. Chained inequalities can be combined to form compound inequalities, such as a < x < b.

Solving Inequalities

Linear Inequalities

Linear inequalities are mathematical expressions that involve a linear relationship between variables. They are represented by an inequality sign rather than an equal sign.

In a single-variable linear inequality, the expression takes the form:

  • ax + b < c
  • ax + b \leq c
  • ax + b > c
  • ax + b \geq c

where a, b, and c are constants and x is the variable.

For example: 3x - 5 > 7

In a two-variable linear inequality, the expression takes the form:

  • ax + by < c
  • ax + by \leq c
  • ax + by > c
  • ax + by \geq c

where a, b, and c are constants and x and y are variables.

For example: 2x + 3y < 6

Solving Simple Linear Inequalities

To solve linear inequalities, follow these steps to isolate the variable:

Step 1: Simplify both sides of the inequality, if necessary, by combining like terms.

Step 2: Use addition or subtraction to eliminate any constant terms on the same side as the variable. For example, to solve the inequality 3x - 5 > 7, first add 5 to both sides:

3x > 12.

Step 3: Use multiplication or division to isolate the variable by eliminating any coefficients. Continuing with the previous example, divide both sides by 3:

x > 4.

When solving linear inequalities, it’s important to remember that the inequality sign should be reversed when multiplying or dividing by a negative number.

For example, to solve -2x \leq 8, divide both sides by -2 and reverse the inequality sign: x \geq -4. This rule exists because multiplying or dividing by a negative number changes the order of the values being compared.

Solving Compound Inequalities

Compound inequalities involve two or more inequalities connected by ‘AND’ or ‘OR’ statements. To solve compound inequalities, solve each inequality separately, following the steps for solving simple linear inequalities.

  • When solving compound inequalities with ‘AND’, find the intersection of the solution sets. This intersection represents the range of values that satisfy both inequalities. For example, to solve x > 2 AND x \leq 5, the solution set is 2 < x \leq 5, which means x is greater than 2 and less than or equal to 5.
  • When solving compound inequalities with ‘OR,’ find the union of the solution sets. The union represents the combined range of values that satisfy either inequality. For example, to solve x < 3 OR x \geq 7, the solution set includes all values of x that are either less than 3 or greater than or equal to 7.

Let’s work through an example:

Solve the compound inequality -3 \leq 2x - 4 < 8

This compound inequality has an “AND” statement because the variable x must satisfy both inequalities simultaneously. To solve the inequality, we’ll isolate x in both inequalities.

Step 1: Split the compound inequality into two separate inequalities.

-3 \leq 2x - 4

2x - 4 < 8

Step 2: Solve the first inequality, -3 \leq 2x - 4.

Add 4 to both sides: -3 + 4 \leq 2x - 4 + 4

Simplify: 1 \leq 2x

Divide by 2: \frac{1}{2} \leq x

Step 3: Solve the second inequality, 2x - 4 < 8

Add 4 to both sides: 2x - 4 + 4 < 8 + 4

Simplify: 2x < 12

Divide by 2: x < 6

Step 4: Combine the inequalities.

\frac{1}{2} \leq x < 6

This is the solution set for the compound inequality, which means that x must be greater than or equal to \frac{1}{2} and less than 6. In interval notation, the solution set can be written as [\frac{1}{2}, 6).

Examples

Let’s look at a few examples to illustrate the concepts discussed above.

Example 1:

Solve the inequality 2x - 6 < 4x + 8.

1. First, subtract 2x from both sides: -6 < 2x + 8-6 < 2x + 8.

2. Then, subtract 8 from both sides: -14 < 2x-14 < 2x.

3. Finally, divide both sides by 2: -7 < x-7 < x, or x > -7x > -7.

The solution set for this inequality is x > -7x > -7.

Example 2:

Solve the compound inequality 3x – 2 < 5 AND x + 1 > 4.

1. Solve the first inequality:

a. Add 2 to both sides: 3x < 73x < 7.

b. Divide by 3: x < \frac{7}{3}x < \frac{7}{3}.

2. Solve the second inequality: Subtract 1 from both sides: x > 3x > 3.

3. Combine the inequalities using ‘AND’: 3 < x < \frac{7}{3}3 < x < \frac{7}{3}.

The solution set for this compound inequality is 3 < x < \frac{7}{3}3 < x < \frac{7}{3}.

You’ve used 10 of your 10 free revision notes for the month

Sign up to get unlimited access to revision notes, quizzes, audio lessons and more

Sign up