Solving Linear Inequalities in One Variable

Introduction

Key Concepts

1. Understanding Linear Inequalities

Linear inequalities in one variable are expressions that compare two linear expressions using inequality symbols such as >, <, ≥, and ≤. Unlike linear equations, which assert equality, linear inequalities express a range of possible solutions. The general form of a linear inequality in one variable is:

where a, b, and c are constants, and x is the variable. The solution to a linear inequality is a set of values for x that make the inequality true.

2. Solving Linear Inequalities

Solving linear inequalities involves finding the range of values for the variable that satisfy the inequality. The process is similar to solving linear equations, with an additional consideration when multiplying or dividing by negative numbers. The key steps are:

1. Isolate the Variable: Use addition or subtraction to move constants to the opposite side of the inequality.
2. Simplify: Combine like terms to simplify the inequality.
3. Solve for the Variable: Divide or multiply both sides by the coefficient of the variable. Remember, if you multiply or divide by a negative number, reverse the inequality symbol.
4. Express the Solution: Write the solution in inequality form, interval notation, or graphically on a number line.

Consider the inequality:

To solve for x:

1. Subtract 3 from both sides: $$ 2x < 4 $$
2. Divide both sides by 2: $$ x < 2 $$

The solution is all real numbers less than 2.

3. Graphical Solutions

Graphing linear inequalities provides a visual representation of the solution set on the number line. To graph:

1. Draw a number line.
2. Plot the boundary point, which is the solution to the corresponding linear equation. For $2x + 3 = 7$, the boundary point is $x = 2$.
3. Use an open circle if the inequality is < or >, indicating that the boundary point is not included.
4. Use a closed circle if the inequality is ≤ or ≥, indicating that the boundary point is included.
5. Shade the region representing the solution set.

Using the previous example, $x < 2$ is graphed by placing an open circle at 2 and shading all numbers to the left.

4. The Impact of Multiplying or Dividing by Negative Numbers

A crucial aspect of solving inequalities is understanding how multiplying or dividing by a negative number affects the inequality symbol. Specifically, doing so reverses the direction of the inequality. For example:

To solve for x:

1. Divide both sides by -3, remembering to reverse the inequality: $$ x < -3 $$

The solution is all real numbers less than -3.

5. Compound Inequalities

Sometimes, inequalities involve two or more inequalities combined. These are known as compound inequalities and can be categorized as:

• Conjunctions (AND): Both inequalities must be true simultaneously. For example: $$ 1 < x + 2 < 5 $$

  Solving gives: $$ -1 < x < 3 $$

• Disjunctions (OR): At least one of the inequalities must be true. For example: $$ x + 2 < 1 \quad \text{OR} \quad x + 2 > 5 $$

  Solving gives: $$ x < -1 \quad \text{OR} \quad x > 3 $$

6. Applications of Linear Inequalities

Linear inequalities are widely used in various real-life contexts, such as:

• Budgeting: Determining the maximum expenditure within a budget constraint.
• Optimizing Resources: Allocating resources to ensure efficiency without exceeding limits.
• Decision Making: Evaluating conditions under which certain decisions are feasible.

For instance, if a student wants to buy notebooks costing $2 each and has $20, the inequality representing this scenario is:

Solving for n: $$ n \leq 10 $$

This indicates the student can purchase up to 10 notebooks.

7. Solving Multi-Step Inequalities

Some inequalities require multiple steps to isolate the variable. Consider:

Solving step-by-step:

1. Expand the left side: $$ 6x - 12 + 5 &geq 2x + 7 $$ $$ 6x - 7 &geq 2x + 7 $$
2. Subtract 2x from both sides: $$ 4x - 7 &geq 7 $$
3. Add 7 to both sides: $$ 4x &geq 14 $$
4. Divide both sides by 4: $$ x &geq 3.5 $$

The solution is $x \geq 3.5$.

8. Checking Solutions

After solving an inequality, it is essential to verify the solution by substituting it back into the original inequality. This ensures the solution satisfies the inequality.

For example, with $x < 2$, substitute $x = 1$:

Substituting $x = 3$:

This confirms that $x < 2$ is the correct solution.

9. Interval Notation and Set Builder Notation

Solutions to inequalities can be expressed in various forms:

• Inequality Form: $x < 2$
• Interval Notation: $(-\infty, 2)$
• Set Builder Notation: $\{x | x < 2\}$

Understanding these notations is crucial for communicating solutions effectively.

10. Compound Inequalities in Real-World Problems

Consider a scenario where a company wants to hire employees who can work between 20 to 40 hours a week. Let x represent the number of hours. The compound inequality representing this requirement is:

This means employees must work at least 20 hours but no more than 40 hours per week.

Comparison Table

Summary and Key Takeaways