Linear inequalities are mathematical expressions involving a linear function and inequality signs (<, >, ≤, and ≥). The process of solving linear inequalities is similar to solving linear equations, but with additional rules when dealing with inequalities.
Understanding Inequalities
An inequality states that one expression is greater or smaller than another. Common inequality symbols include:
- < (Less than)
- > (Greater than)
- ≤ (Less than or equal to)
- ≥ (Greater than or equal to)
For example, the inequality 3x + 5 < 11 means that the value of x must be such that when multiplied by 3 and increased by 5, the result is less than 11.
Solving Linear Inequalities
To solve an inequality, follow similar steps as solving an equation while keeping in mind one crucial rule:
The inequality sign should be inverted when dividing or multiplying by a negative value.
Example 1:
Solve: 2x + 3 ≤ 11
Step 1: Subtract 3 from both sides:
2x ≤ 8
Step 2: Divide by 2:
x ≤ 4x
Thus, the solution is x ≤ 4.
Example 2 (Reversing the Inequality Sign):
Solve: -3x + 7 > 1
Step 1: Subtract 7 from both sides:
−3x > −6
Step 2: Divide by -3 (flip the sign!):
x < 2x
So, the solution is x < 2.
Representing Solutions on a Number Line
- Use an open circle (○) for < or > (not including the value).
- Use a closed circle (●) for ≤ or ≥ (including the value).
For example, x ≤ 4 is represented with a closed circle at 4 and shading to the left.
Solving Compound Inequalities
A compound inequality involves two inequalities, such as:
−2 ≤ 3x − 1 < 8
Step 1: Add 1 to all parts:
−1 ≤ 3x < 9
Step 2: Divide by 3:
−13 ≤ x < 3
Solution: −13 ≤ x < 3
Conclusion
Solving linear inequalities is crucial in algebra. Always remember to reverse the inequality sign when multiplying or dividing by a negative number. Represent solutions clearly on a number line and apply logical steps to solve inequalities correctly.