Problem 78 - Beginning and Intermediate Algebra

Question

The following exercises contain absolute value equations, linear inequalities, and both types of absolute value inequalities. Solve each. Write the solution set for equations in set notation and use interval notation for inequalities. $$|7-x| \leq 12$$

Step-by-Step Solution

Answer

To solve the given absolute value inequality \(|7-x| \leq 12\), we can rewrite it as \(-12 \leq (7-x) \leq 12\). This results in two linear inequalities: \(7-x \geq -12\) and \(7-x \leq 12\). Solving these inequalities individually, we get \(x \leq 19\) and \(x \geq -5\). Combining the inequalities, we have \(-5 \leq x \leq 19\), which can be written in interval notation as \([-5, 19]\).

Step 1: Understand absolute value inequality properties

An absolute value inequality of the form \(|a| \leq b\) for real numbers a and b can be represented as \(-b \leq a \leq b\). So, our given inequality can be written as: $$-12 \leq (7-x) \leq 12$$

Step 2: Solve the two linear inequalities

The inequality above consists of two linear inequalities: 1. \(7 - x \geq -12\) 2. \(7 - x \leq 12\)

Step 3: Solve the first linear inequality \(7-x \geq -12\)

To solve this inequality, isolate x by performing the following steps: 1. Subtract 7 from both sides: \(-x \geq -19\) 2. Multiply both sides by -1 to eliminate the negative sign of x (note that this reverses the inequality sign): \(x \leq 19\)

Step 4: Solve the second linear inequality \(7 - x \leq 12\)

To solve this inequality, also isolate x: 1. Subtract 7 from both sides: \(-x \leq 5\) 2. Multiply both sides by -1 (note that this reverses the inequality sign): \(x \geq -5\)

Step 5: Combine the inequalities and write the solution in interval notation

Now we combine the inequalities from Step 3 and Step 4: $$-5 \leq x \leq 19$$ This is our final solution in interval notation: \([-5, 19]\).

Key Concepts

Linear Inequalities

Linear inequalities are a type of mathematical expression used to determine the range of possible values for a variable, usually denoted as "x." When solving linear inequalities, we deal with expressions that involve inequalities such as "less than," "greater than," "less than or equal to," and "greater than or equal to." For example, in our case regarding the inequality \( |7 - x| \leq 12 \), it translates to two different linear inequalities:

• \( 7 - x \geq -12 \)
• \( 7 - x \leq 12 \)

Linear inequalities are solved by isolating the variable on one side. Often, this process includes adding or subtracting terms and sometimes multiplying or dividing both sides of the inequality by a number, which can reverse the inequality sign. Understanding these steps thoroughly ensures you can handle any linear inequality challenge.

Set Notation

Set notation is a mathematical language used to describe a set of solutions, or collection of elements that satisfy certain conditions. In the context of inequalities, set notation can express the solutions to equations or inequalities based on the conditions specified. For instance, if you have an equation like \( x^2 = 4 \), you can express the solution set in set notation as \( \{ x | x = -2 \text{ or } x = 2 \} \). However, for inequalities, such as those resulting from the problem \( |7 - x| \leq 12 \), interval notation is more commonly used for solution sets, especially if the set includes an infinite number of solutions.

Interval Notation

Interval notation is a concise way of representing a range of values in mathematics. For inequalities, it indicates the span of solutions that satisfy the inequality. The use of brackets and parentheses in interval notation communicates whether end values are included or excluded in the solution set:

• Brackets [ ] are used when end values are included (representing "less than or equal to" or "greater than or equal to").
• Parentheses ( ) are used when end values are not included (representing "less than" or "greater than").

For the inequality \( -5 \leq x \leq 19 \), interval notation represents it as \([-5, 19]\), indicating that \( x \) can be any value from -5 to 19, inclusive. This notation simplifies the way we communicate the solution to an inequality.

Solving Inequalities

Solving inequalities involves finding the set of values of a variable that satisfies the inequality condition. The process often resembles solving equations but comes with an important caveat: when multiplying or dividing both sides of an inequality by a negative number, the direction of the inequality sign must be reversed to maintain a true statement. This reversal ensures that the inequality's balance is preserved. In the case of the inequality \( |7 - x| \leq 12 \), we first break it down into two separate inequalities:

• \( 7 - x \geq -12 \): Solving this gives us \( x \leq 19 \) after manipulating the terms.
• \( 7 - x \leq 12 \): Solving this gives us \( x \geq -5 \).

By combining these two inequalities, we arrive at \( -5 \leq x \leq 19 \), which is the solution. Understanding these steps ensures clarity in arriving at the right solution for any inequality problem.