Question

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

Short answer

\( x > x + 1 \) has no solution. \( (x - 2)^2 = 0 \) has exactly one solution.

Step-by-step solution

- Inequality with No Solution

An inequality with no solution is one where no value for the variable makes the inequality true. For example, consider the inequality \( x > x + 1 \). There is no value for \( x \) that can satisfy this because a number is never greater than itself plus 1.

- Inequality with Exactly One Solution

An inequality with exactly one solution occurs when there's only one specific value of the variable that satisfies the inequality. For example, consider \( (x - 2)^2 = 0 \). Rewrite it as \( x - 2 = 0 \). Solving this gives us \( x = 2 \), which is the only solution.

Key concepts

No Solution Inequality

In mathematics, an inequality that has no solution is one where no possible value for the variable will make the inequality true.
For example, let's look at the inequality \( x > x + 1 \).
This inequality states that a number \( x \) should be greater than itself plus 1, which is impossible.
How can \( x \) be greater than \( x + 1 \)? It can't, because adding 1 to \( x \) means it will always be greater than \( x \).
Therefore, no value of \( x \) can satisfy this condition, which makes this inequality an excellent example of one with no solution.

Single Solution Inequality

A single solution inequality is one where just one specific value of the variable makes the inequality true.
Consider the example \( (x - 2)^2 = 0 \).
First, rewrite it as \( x - 2 = 0 \).
Now, solve for \( x \) by adding 2 to both sides, resulting in \( x = 2 \).
This tells us that when \( x \) equals 2, the inequality holds true.
Only this single value works because any other value will not satisfy the equation, making this a perfect illustration of a single solution inequality.

Solving Inequalities

Solving inequalities involves finding the values of a variable that make the inequality true.
Here are a few key points to keep in mind:

• Always perform the same operation on both sides of the inequality to maintain balance.
• If you multiply or divide by a negative number, you must reverse the inequality sign.
• Check your solution by substituting values back into the original inequality to verify they work.

For example, let's solve \( 3x + 5 > 11 \):

• Subtract 5 from both sides: \( 3x > 6 \).
• Divide both sides by 3: \( x > 2 \).

So, any value greater than 2 will satisfy the inequality.
Just remember these basic steps, and solving inequalities will feel much simpler!