Problem 24 - Algebra and Trigonometry

Question

Solve the inequality. Express the answer using interval notation. $$ |x-9|>9 $$

Step-by-Step Solution

Answer

The solution is \((-\infty, 0) \cup (18, \infty)\).

Step 1: Understand the Absolute Value Inequality

The inequality \(|x-9| > 9\) implies that the distance of \(x\) from 9 is greater than 9. This means \(x\) could be either more than 9 units away positively or more than 9 units away negatively.

Step 2: Break Into Two Inequalities

Since \(|x-9| > 9\), we can split this into two separate inequalities: 1. \(x - 9 > 9\) 2. \(x - 9 < -9\).

Step 3: Solve the First Inequality

For the inequality \(x - 9 > 9\), solve for \(x\):\[ x - 9 > 9 \]Add 9 to both sides:\[ x > 18 \].

Step 4: Solve the Second Inequality

For the inequality \(x - 9 < -9\), solve for \(x\):\[ x - 9 < -9 \]Add 9 to both sides:\[ x < 0 \].

Step 5: Combine Solutions Using Interval Notation

The solutions to the inequalities \(x > 18\) and \(x < 0\) are combined using the union of two intervals:\((-\infty, 0) \cup (18, \infty)\).

Key Concepts

Interval Notation

Interval notation is a shorthand used in mathematics to show the set of all solutions to an inequality. This notation uses brackets and parentheses to depict ranges of numbers without listing every individual solution.

• Use parentheses, \((\), when a number is not included in the set. For example, \((0, 1)\) includes all the numbers between 0 and 1 but not 0 and 1 themselves.
• Use brackets, \([\), when a number is included in the set. For example, \([0, 1]\) includes 0, 1, and all numbers between them.

In the original exercise, the solution is expressed as \((-\infty, 0) \cup (18, \infty)\). This means that \(x\) can be less than 0 or more than 18, but not exactly 0 or 18. Using interval notation is straightforward once you understand these basic rules and it offers a clear and concise way to express solutions.

Solving Inequalities

Solving inequalities involves finding the set of numbers that make the inequality true. Inequalities like \(|x-9|>9\) describe a range of answers rather than just one solution. To solve absolute value inequalities, the process often involves splitting the absolute value expression into two separate inequalities. For \(|x-9|>9\), think of it as a boundary problem, where you're finding when the expression inside the absolute value is further away from zero than 9. This splits into two cases:

• One where \((x - 9) > 9\)
• Another where \((x - 9) < -9\)

By solving each inequality separately:

• \(x - 9 > 9\) simplifies to \(x > 18\) by adding 9 to both sides.
• \(x - 9 < -9\) simplifies to \(x < 0\) by adding 9 to both sides.

Once you solve each part, you combine the answers, using an 'OR' approach, because the solution set includes either inequality.

Distance on the Number Line

Understanding absolute value in terms of distance on the number line can simplify many problems.

• The absolute value \(|x-9|\) represents the distance of \(x\) from 9.
• If \(|x-9| > 9\), it means that \(x\) is more than 9 units away from the point 9 either to the left or right on the number line.

Visualizing this on a number line can be very helpful:- Numbers greater than 18 lie to the right of 9 and handle part of the inequality \((x > 18)\).- Numbers less than 0 lie to the left of 9 and handle the other part \((x < 0)\).This visualization helps grasp how the solutions of absolute value inequalities form part of segments on the number line rather than just single points. It's as if we're creating ranges in a treasure map, where the 'treasure' is the spot where the inequality is true, spread over a wide territory.