In mathematics, an inequality compares two values, showing if one is less than, greater than, less than or equal to, or greater than or equal to another. Checking solutions to inequalities involves substituting a given value for the variable and determining if the resulting statement is true. If the statement is true, the value is a solution; otherwise, it is not.
For example, consider the inequality $x + 3 > 5$. If we substitute $x = 3$, we get $3 + 3 > 5$, which simplifies to $6 > 5$. This statement is true, so $x = 3$ is a solution. If we substitute $x = 1$, we get $1 + 3 > 5$, which simplifies to $4 > 5$. This statement is false, so $x = 1$ is not a solution.
Match each term with its definition:
| Term | Definition |
|---|---|
| 1. Inequality | A. A value that makes the inequality true. |
| 2. Solution | B. A symbol indicating less than or equal to. |
| 3. Greater Than | C. A mathematical statement comparing two expressions using symbols like >, <, $\geq$, or $\leq$. |
| 4. Less Than or Equal To | D. A symbol indicating that one value is larger than another. |
| 5. Variable | E. A symbol representing an unknown value. |
An _________ is a statement that compares two expressions using symbols like >, <, $\geq$, or $\leq$. A _________ to an inequality is a value that makes the inequality true. When checking a potential solution, _________ the value for the variable and see if the resulting statement is _________. If it is, the value is a solution; if not, it is not. The symbols $\geq$ and $\leq$ mean 'greater than or equal to' and '_________', respectively.
Explain, in your own words, why it's important to check solutions when working with inequalities. Give an example to illustrate your point.
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