Question

Make up an inequality that has no solution. Make up one that has exactly one solution.

Short answer

Examples: 1. \[ x + 3 < x + 1 \]— No solution.2. \[ 2(x - 1) = x + 1 \]— Exactly one solution: \[ x = 3 \].

Step-by-step solution

Inequality with No Solution

One way to create an inequality with no solution is to present conflicting conditions. For example, consider the inequality \[ x + 3 < x + 1 \].We'll simplify this:

Simplifying the Inequality

Subtract x from both sides: \[ x + 3 - x < x + 1 - x \]This reduces to: \[ 3 < 1 \]Since 3 is never less than 1, this inequality has no solution.

Inequality with Exactly One Solution

Create an inequality where solving it results in a single value for x. For example, consider: \[ 2(x - 1) = x + 1 \].

Solving the Inequality

First, expand and move all terms involving x to one side:\[ 2x - 2 = x + 1 \]Subtract x from both sides:\[ x - 2 = 1 \]Add 2 to both sides:\[ x = 3 \]Thus, the inequality has exactly one solution: \[ x = 3 \].

Key concepts

inequalities with no solution

Inequalities can sometimes have no solutions. This happens when the conditions set by the inequality cannot ever be satisfied. For example, consider the inequality \[x + 3 < x + 1\].
This statement implies that 3 is less than 1, which is never true. When simplifying, you subtract x from both sides, resulting in \[3 < 1\].

Because this inequality will never hold true, there is no value for x that can satisfy it. Hence, it is an example of an inequality with no solution.
Understanding such cases helps avoid futile attempts at finding solutions.

solving linear inequalities

Solving linear inequalities involves isolating the variable on one side of the inequality sign. Let’s consider the inequality \[2(x - 1) = x + 1\].

First, expand the left-hand side: \[2x - 2\]. Next, subtract x from both sides: \[2x - 2 - x = x + 1 - x\], giving \[x - 2 = 1\].
Finally, add 2 to both sides to solve for x: \[x = 3\].

This method involves simple arithmetic steps:

• Expand expressions.
• Move terms involving x to one side.
• Isolate x to find its value.

By following these steps, you can solve most linear inequalities efficiently.

single-variable inequalities

Single-variable inequalities involve expressions with only one variable. These are simpler to solve and understand. For example, if the inequality is \[x + 3 < x + 1\], you can subtract x from both sides:
\[ x + 3 - x < x + 1 - x \], yielding \[3 < 1\].

Since 3 is never less than 1, this inequality has no solution. Single-variable inequalities help build a foundation for understanding more complex multi-variable cases.
Common techniques in solving such inequalities include:

• Combining like terms.
• Isolating the variable.
• Applying inverse operations (addition, subtraction, multiplication, or division).

Practice consistently to get comfortable with these basic steps.