First, solve the inequality:
["# How to Solve Inequalities: A Step-by-Step Guide", "Analyzing and solving inequalities is a fundamental skill in algebra that plays a crucial role in mathematics, science, engineering, economics, and everyday decision-making. Whether you're balancing budgets, measuring risks, or optimizing data models, understanding how to solve inequalities is essential. In this article, we’ll walk you through the process of solving inequalities step by step — starting with the basics and moving toward more complex scenarios.", "In this article, we begin by defining what an inequality is and then explore a methodical approach to solving them, using clear examples along the way. By mastering these techniques, you’ll be well-equipped to tackle real-world problems involving unknown values constrained by limits.", "---", "## What Is an Inequality?", "An inequality expresses the relationship between two expressions, indicating whether one is less than, greater than, less than or equal to, or greater than or equal to the other. Common types include:", "- Simple linear inequalities: such as (2x + 3 < 7)\n- Composite inequalities: like ( -4 < 3x - 2 \leq 8 )\n- Absolute value inequalities: for example ( |x - 1| < 5 )", "Inequalities differ from equations because they define a range of possible solutions rather than a single value. Solving an inequality means finding all values of the variable that satisfy the relationship.", "---", "## Step 1: Rewrite the Inequality Without Fractions and Parentheses", "When starting with a linear inequality, clear the equation of fractions and simplify both sides. For example:", "[\n\frac{2x - 5}{3} > 4\n]", "Multiply both sides by 3 (a positive number), preserving the inequality direction:", "[\n2x - 5 > 12\n]", "---", "## Step 2: Isolate the Variable", "Now, solve for ( x ) using inverse operations. Continue simplifying:", "[\n2x > 12 + 5\n\Rightarrow 2x > 17\n]", "Divide both sides by 2:", "[\nx > \frac{17}{2} \quad \ ext{or} \quad x > 8.5\n]", "This inequality states that ( x ) can be any number greater than 8.5.", "---", "## Step 3: Represent the Solution Set on a Number Line", "Visual tools help interpret solutions. On a number line, the solution ( x > 8.5 ) is shown with an open parenthesis at 8.5 (since 8.5 is not included) and shading to the right indicating all larger numbers.", "---", "## Step 4: Solve Composite Inequalities", "Composite inequalities combine two inequalities. Consider:", "[\n-6 \leq 2x + 1 < 9\n]", "This means two conditions:", "1. ( 2x + 1 \geq -6 )\n2. ( 2x + 1 < 9 )", "Solve each separately.", "First inequality:\n[\n2x + 1 \geq -6\n\Rightarrow 2x \geq -7\n\Rightarrow x \geq -\frac{7}{2} \quad \ ext{(or } x \geq -3.5\ ext{)}\n]", "Second inequality:\n[\n2x + 1 < 9\n\Rightarrow 2x < 8\n\Rightarrow x < 4\n]", "Combine both results:", "[\n-\frac{7}{2} \leq x < 4\n]", "This interval shows ( x ) is between (-3.5) and (4), inclusive on the left, exclusive on the right.", "---", "## Step 5: Handle Absolute Value Inequalities", "Absolute value inequalities use interval logic based on definition of distance. Remember:", "[\n|x - a| < b \quad \Rightarrow \quad a - b < x < a + b \quad (b > 0)\n]\n[\n|x - a| > b \quad \Rightarrow \quad x < a - b \quad \ ext{or} \quad x > a + b \quad (b > 0)\n]", "Example:\n[\n|2x - 3| < 7\n]", "This becomes:", "[\n-7 < 2x - 3 < 7\n]", "Solve each part:\n( -7 < 2x - 3 \Rightarrow -4 < 2x \Rightarrow x > -2 )\n( 2x - 3 < 7 \Rightarrow 2x < 10 \Rightarrow x < 5 )", "So, the solution is:\n[\n-2 < x < 5\n]", "---", "## Step 6: Special Cases and Negative Coefficients", "When multiplying or dividing both sides of an inequality by a negative number, reverse the inequality sign.", "Example:\n[\n-3x \geq 12\n\Rightarrow x \leq -4 \quad \ ext{(note: inequality flips to ≤)}\n]", "---", "## Step 7: Real-World Applications", "Solving inequalities isn't just academic — it applies directly to:", "- Budgeting: Finding income levels that satisfy spending constraints.\n- Engineering: Ensuring material limits stay within safe bounds.\n- Optimization: Maximizing profit or minimizing cost under variable limits.\n- Data science: Interpreting confidence intervals and trends.", "Imagine setting a safe speed limit region:\nIf ( 50 < v \leq 70 ), your vehicle must go faster than 50 km/h but not exceed 70.", "---", "## Summary", "Solving inequalities involves:", "1. Simplifying both sides\n2. Isolating the variable using inverse operations\n3. Checking the direction of the inequality after multiplication/division by negatives\n4. Expressing the solution clearly on a number line or interval notation\n5. Applying logic to compound, absolute value, and real-world problems", "Mastering these techniques empowers you to reason precisely about unknowns constrained by real limits — an indispensable tool across disciplines.", "---", "### Quick Recap: Common Inequality Forms\n| Form | Solution Interval Expression |\n|--------------------------------|----------------------------------------|\n| ( ax + b < c ) | ( x < \frac{c - b}{a} ) (if ( a > 0 )) |\n| ( ax + b \geq c ) | ( x \geq \frac{c - b}{a} ) |\n| ( |x - a| < b ) | ( a - b < x < a + b ) |\n| ( |x - a| > b ) | ( x < a - b ) or ( x > a + b ) |", "---", "Whether you're a student, teacher, or professional, sharpening your inequality-solving skills opens doors to clearer, more confident decision-making in mathematical and practical domains. Start practicing today — each inequality solved brings you one step closer to mastery.", "---", "Further Reading & Resources:\n- Khan Academy: Inequalities (Free video lessons)\n- Paul’s Online Math Notes: Inequalities chapter\n- Interactive inequality solvers and graphing tools online", "午夜提醒: Understanding inequalities builds analytical thinking — use it daily!"]
