Problem 48 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 21-70: Solve the inequality, and express the solutions in terms of intervals whenever possible. $$ |x| \leq 7 $$

Step-by-Step Solution

Answer

The solution is the interval \([-7, 7]\).

Step 1: Understand absolute value property

The absolute value \( |x| \) generates two potential inequalities. The inequality \( |x| \leq 7 \) represents all numbers whose distance from zero is less than or equal to 7 on the number line. Therefore, this becomes \( -7 \leq x \leq 7 \).

Step 2: Solve the compound inequality

The inequality \( -7 \leq x \leq 7 \) is a compound inequality. This means that \( x \) is greater than or equal to \(-7\) and less than or equal to \(7\).

Step 3: Express the solution in interval notation

Write the solution for the inequality \( -7 \leq x \leq 7 \) in interval notation. The solution is represented as the interval \([-7, 7]\), which includes all values between \(-7\) and \(7\), inclusive of both endpoints.

Key Concepts

Compound Inequalities

Compound inequalities are a way to express multiple inequalities at once. They are two or more inequalities that are joined by the word "and" or "or", and they describe a range of numbers that satisfy all parts of the inequality simultaneously. In our exercise, the absolute value inequality \(|x| \leq 7\) converts into the compound inequality \(-7 \leq x \leq 7\).

• The first part of the inequality is \(x \geq -7\), meaning that \(x\) can be any number greater than or equal to \(-7\).
• The second part is \(x \leq 7\), indicating \(x\) can be any number less than or equal to \(7\).

These two conditions together ensure that \(x\) lies within the range from \(-7\) to \(7\). The solution forms a balanced statement, ensuring \(x\) meets both criteria simultaneously.

Interval Notation

Interval notation is a shorthand used in mathematics to represent the solutions of inequalities. It's a concise way to show where an inequality's solution lies on a number line. Instead of writing out the entire inequality, interval notation uses brackets to show the endpoints of the range of solutions.

• Use square brackets \([ \ ])\) when the endpoints are included in the solution.
• Use parentheses \(( \ ))\) when the endpoints are not included.

In the inequality \(-7 \leq x \leq 7\), both endpoints \(-7\) and \(7\) are included, as indicated by the "less than or equal to" symbol. This means \(x\) can be \(-7\) or \(7\), along with any number in between. Consequently, it is written in interval notation as \([-7, 7]\). This format succinctly communicates that every number between \(-7\) and \(7\), including the endpoints, is part of the solution set. Interval notation is especially helpful for quickly visualizing and working with ranges of numbers.

Solving Inequalities

Solving inequalities involves finding the set of all values that make the inequality true. Unlike equations, inequalities include a range of possible answers instead of one specific solution. When you solve inequalities, you're determining which values satisfy the condition described by the inequality.To solve an absolute value inequality such as \(|x| \leq 7\), first understand what absolute value represents. It signifies the distance a number is from zero without considering direction. Therefore, \(|x| \leq 7\) means \(x\) is within 7 units of zero. From this, establish the compound inequality \(-7 \leq x \leq 7\), which includes analyzing:

• What happens if \(x\) is negative, hence \(x \geq -7\).
• What happens if \(x\) is positive, resulting in \(x \leq 7\).

After forming these inequalities, check the range by testing values within and outside it to verify they fit the original inequality. Solving inequalities provides insight into how variables behave across a continuum, as opposed to fixed points.