Sure! Let's break this down step-by-step to correctly match each inequality notation with its corresponding interval notation.
1. Inequality: [tex]\(x > 7.8\)[/tex]
- This means [tex]\(x\)[/tex] is greater than 7.8 but not equal to 7.8.
- In interval notation, this would be represented as [tex]\((7.8, \infty)\)[/tex], since the interval starts just above 7.8 and extends to positive infinity.
2. Inequality: [tex]\(x \geq 7.8\)[/tex]
- This means [tex]\(x\)[/tex] is greater than or equal to 7.8.
- In interval notation, this would be represented as [tex]\([7.8, \infty)\)[/tex], since the interval includes 7.8 and extends to positive infinity.
3. Inequality: [tex]\(x < 7.8\)[/tex]
- This means [tex]\(x\)[/tex] is less than 7.8 but not equal to 7.8.
- In interval notation, this would be represented as [tex]\((-\infty, 7.8)\)[/tex], since the interval starts from negative infinity and goes up to, but does not include, 7.8.
4. Inequality: [tex]\(x \leq 7.8\)[/tex]
- This means [tex]\(x\)[/tex] is less than or equal to 7.8.
- In interval notation, this would be represented as [tex]\((-\infty, 7.8]\)[/tex], since the interval starts from negative infinity and includes 7.8.
Let's match these correctly as per the solution set:
- [tex]\(x > 7.8\)[/tex] ↔ [tex]\((7.8, \infty)\)[/tex]
- [tex]\(x \geq 7.8\)[/tex] ↔ [tex]\([7.8, \infty)\)[/tex]
- [tex]\(x < 7.8\)[/tex] ↔ [tex]\((-\infty, 7.8)\)[/tex]
- [tex]\(x \leq 7.8\)[/tex] ↔ [tex]\((-\infty, 7.8]\)[/tex]
By placing the correct matches next to the inequalities, we get:
[tex]\[
\begin{array}{ll}
x \geq 7.8 & \rightarrow [7.8, \infty) \\
x > 7.8 & \rightarrow (7.8, \infty) \\
x < 7.8 & \rightarrow (-\infty, 7.8) \\
x \leq 7.8 & \rightarrow (-\infty, 7.8]
\end{array}
\][/tex]
These are the correct matches for the inequality sets with their respective interval notation.
