1. Let's start by stating the problem: A mixed inequality involves two inequalities joined together, such as $a < x \leq b$. It means $x$ is a number that satisfies both inequalities simultaneously. 2. The general form of a mixed inequality is: $$a < x < b$$ or $$a \leq x \leq b$$ where $a$ and $b$ are constants, and $x$ is the variable. 3. Important rules: - The variable $x$ must satisfy both inequalities at the same time. - The inequality signs indicate whether the endpoints $a$ and $b$ are included ($\leq$ or $\geq$) or excluded ($<$ or $>$). 4. To solve or interpret a mixed inequality, you find all values of $x$ that lie between $a$ and $b$ according to the inequality signs. 5. Example: Solve $2 < x \leq 5$. - This means $x$ is greater than 2 but less than or equal to 5. - So the solution set is all $x$ such that $2 < x \leq 5$. 6. In interval notation, this is written as $(2, 5]$. 7. Graphically, this is represented on a number line with an open circle at 2 (not included) and a closed circle at 5 (included), shading all points in between. 8. Mixed inequalities are useful for expressing ranges of values and are common in algebra and calculus. Final answer: A mixed inequality like $a < x \leq b$ means $x$ lies between $a$ and $b$, including or excluding endpoints based on the inequality signs.