Problem 16 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether each value of \(x\) is a solution of the inequality. Inequality $$|x-10| \geq 3$$ Values (a) \(x=13\) (b) \(x=-1\) (c) \(x=14\) (d) \(x=8\)

Step-by-Step Solution

Answer

The solutions to the given inequality \(|x-10| \geq 3\) are \(x=13\), \(x=-1\), and \(x=14\). \(x=8\) is not a solution.

Step 1: Check Value (a)

Substitute \(x=13\) into the inequality. We get \(|13 - 10| \geq 3\), which simplifies to \(3 \geq 3\). Since 3 is equal to 3, \(x=13\) is a solution to the inequality.

Step 2: Check Value (b)

Substitute \(x=-1\) into the inequality. We get \(|-1 - 10| \geq 3\), which simplifies to \(11 \geq 3\). Since 11 is greater than 3, \(x=-1\) is a solution to the inequality.

Step 3: Check Value (c)

Substitute \(x=14\) into the inequality. We get \(|14 - 10| \geq 3\), which simplifies to \(4 \geq 3\). Since 4 is greater than 3, \(x=14\) is a solution to the inequality.

Step 4: Check Value (d)

Substitute \(x=8\) into the inequality. We get \(|8 - 10| \geq 3\), which simplifies to \(2 \geq 3\). Since 2 is not greater than or equal to 3, \(x=8\) is not a solution to the inequality.

Key Concepts

Absolute Value

The absolute value of a number is an important concept in algebra that represents the distance of a number from zero on the number line. It is always expressed as a non-negative value, regardless of whether the original number is positive or negative. For example, the absolute value of both -5 and 5 is 5. To notate absolute value, we use vertical bars on either side of the number, like this:

• \(|x|\) means "the absolute value of x."

The key idea here is to remember that the absolute value focuses on magnitude, not direction. So, when working with absolute value expressions like \(|x-10|\), it's about finding how far \(x\) is from 10, regardless of whether it's greater than (right of) or less than (left of) 10 on the number line.

Inequalities

Inequalities are used extensively in algebra to express how one value relates to another. They show that expressions are not always equal, but instead involve a range of potential solutions. Inequalities are represented with symbols that show less than, greater than, less than or equal to, or greater than or equal to a given value. Here are the symbols commonly used in inequalities:

• \(<\) means "less than."
• \(>\) means "greater than."
• \(\leq\) means "less than or equal to."
• \(\geq\) means "greater than or equal to."

When solving inequalities, it's important to remember that the direction of the inequality symbol must remain intact unless multiplication or division by a negative number occurs. For example, in an expression like \(|x-10| \geq 3\), we are looking at values of \(x\) that keep the absolute difference from 10 at least 3 units away, whether greater or less.

Solution to Inequalities

Solving inequalities is about finding all possible values that make the inequality true. The process involves evaluating whether a particular value satisfies the inequality, similar to how you would test for equality, but with the added consideration of the inequality symbols. To check if a value is a solution to the inequality \(|x-10| \geq 3\):

• Substitute the value into the inequality.
• Simplify the expression to evaluate the absolute value.
• Check if the simplified expression satisfies the inequality condition.

In our exercise, each value of \(x\) is checked separately:

• For \(x = 13\), \, \(|13 - 10| \) simplifies to 3, which satisfies \( 3 \geq 3\).
• For \(x = -1\), \, \(|-1 - 10| \) simplifies to 11, which is greater than 3.
• For \(x = 14\), \, \(|14 - 10|\) simplifies to 4, satisfying the condition \(4 \geq 3\).
• For \(x = 8\), \, \(|8 - 10| \) simplifies to 2, which does not satisfy the inequality \(2 \geq 3\).

Thus, solving inequalities involves understanding absolute differences and relationships defined by inequality symbols, leading to identifying the valid solutions.