How many solutions does a linear inequality have?

A linear inequality has either infinitely many solutions or no solution.

Understanding Linear Inequalities

Linear inequalities, unlike linear equations, do not have a single, definitive answer. Instead, they define a range of values that satisfy the inequality.

• Infinitely Many Solutions: This is the most common scenario. The solution set encompasses all numbers that, when substituted into the inequality, make the statement true. These solutions can be represented on a number line or using interval notation.

  • Example: x > 3. Any number greater than 3 is a solution (e.g., 3.0001, 4, 100, 1000000). This inequality has infinitely many solutions. The solution can be represented on a number line by shading all numbers to the right of 3 (with an open circle at 3 to indicate that 3 itself is not included). In interval notation, it is expressed as (3, ∞).

• No Solution: In certain cases, no value will satisfy the inequality.

  • Example: x + 1 < x. If you subtract x from both sides, you get 1 < 0, which is always false. Therefore, this inequality has no solution.

Representation of Solutions

When a linear inequality has infinitely many solutions, we represent these solutions in the following ways:

• Number Line: A number line is used to visually represent the range of values that satisfy the inequality. An open circle indicates that the endpoint is not included in the solution (e.g., for x > 2), while a closed circle indicates that the endpoint is included (e.g., for x ≥ 2).

• Interval Notation: Interval notation uses parentheses and brackets to represent the range of solutions. Parentheses ( ) indicate that the endpoint is not included, while brackets [ ] indicate that it is included. Infinity is always represented with a parenthesis.

  • x > 5 is represented as (5, ∞).
  • x ≤ -2 is represented as (-∞, -2].
  • -1 < x ≤ 4 is represented as (-1, 4].

Summary Table

Inequality	Number of Solutions	Example	Solution Representation
x > a	Infinitely many	x > 2	Number line, (2, ∞)
x ≥ a	Infinitely many	x ≥ -1	Number line, [-1, ∞)
x < a	Infinitely many	x < 7	Number line, (-∞, 7)
x ≤ a	Infinitely many	x ≤ 0	Number line, (-∞, 0]
Always False (e.g., 1 < 0)	No solution	x + 1 < x	N/A

In conclusion, linear inequalities generally have infinitely many solutions, but can also have no solution depending on the specific inequality.