Linear inequalities are mathematical expressions that compare two values using inequality symbols like < (less than), > (greater than), $\leq$ (less than or equal to), and $\geq$ (greater than or equal to). When solving word problems involving linear inequalities, it's crucial to translate the given information into a mathematical statement. Identify keywords that indicate inequalities, such as 'at least,' 'no more than,' 'minimum,' or 'maximum.' Once you set up the inequality, solve it using algebraic techniques similar to solving equations, but remember that multiplying or dividing by a negative number reverses the inequality sign.
For example, consider the statement "Five times a number plus two is less than or equal to seventeen." This translates to the inequality $5x + 2 \leq 17$. Solving for $x$ involves subtracting 2 from both sides ($5x \leq 15$) and then dividing by 5 ($x \leq 3$). The solution indicates that any number less than or equal to 3 satisfies the original statement.
Match the term with its definition:
| Term | Definition |
|---|---|
| 1. Inequality | A. A value that, when substituted for a variable, makes the inequality true. |
| 2. Variable | B. A mathematical statement that compares two expressions using inequality symbols. |
| 3. Solution Set | C. A symbol representing an unknown quantity. |
| 4. Coefficient | D. A number multiplied by a variable in an algebraic expression. |
| 5. Constant | E. A fixed value that does not change. |
Complete the following paragraph using the words: inequality, greater, less, solution, variable.
A linear __________ is a statement that compares two expressions. The symbol '>' means __________ than, while '<' means __________ than. Solving an inequality means finding the __________ set, which includes all values of the __________ that make the inequality true.
Explain, in your own words, how solving a linear inequality is similar to and different from solving a linear equation.
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