Problem 13 - Algebra for College Students

Question

Solve each of the inequalities and express the solution sets in interval notation. \(0.06 x+0.08(250-x) \geq 19\)

Step-by-Step Solution

Answer

The solution set is \((-\infty, 50]\) in interval notation.

Step 1: Simplify the Inequality

Begin by distributing 0.08 into the expression inside the parentheses.The inequality is:\[0.06x + 0.08(250 - x) \geq 19\]Distribute 0.08:\[0.06x + 20 - 0.08x \geq 19\]

Step 2: Combine Like Terms

Combine the terms with \(x\) in them.\[0.06x + 20 - 0.08x \geq 19\]Simplify:\[-0.02x + 20 \geq 19\]

Step 3: Isolate the Variable Term

Subtract 20 from both sides to isolate the term with \(x\).\[-0.02x \geq 19 - 20\]\[-0.02x \geq -1\]

Step 4: Solve for x

Divide both sides by \(-0.02\). Remember that dividing by a negative number reverses the inequality sign.\[x \leq \frac{-1}{-0.02}\]\[x \leq 50\]

Step 5: Express in Interval Notation

The solution \(x \leq 50\) in interval notation is \(( -\infty, 50 ]\).

Key Concepts

Algebraic Expressions

Understanding algebraic expressions is key to solving inequalities. Algebraic expressions consist of variables, constants, and operations like addition, subtraction, multiplication, or division. In our inequality example, the expression \(0.06x + 0.08(250 - x)\) includes:

• A variable: \(x\), which acts as a placeholder that can have various values.
• Coefficients: \(0.06\) and \(0.08\), which multiply the variable or parts of the expression.
• A constant: \(250\), which doesn't change.

These parts come together to form the algebraic expression. When solving inequalities, we manipulate these expressions following mathematical properties and rules to find the range of \(x\) that satisfies the inequality. Recognizing these components can help simplify and solve complex mathematical problems.

Interval Notation

Interval notation is a method of writing sets of numbers between two endpoints. It's particularly useful when expressing solutions to inequalities. In the context of inequalities, like the one solved here, we focus on the range of values that satisfies the inequality.

• An open interval, such as \((a, b)\), doesn't include its endpoints and represents all numbers between \(a\) and \(b\).
• A closed interval, like \([a, b]\), includes its endpoints and represents all numbers between \(a\) and \(b\), inclusive.
• A half-open interval, such as \((a, b]\), includes \(b\) but not \(a\), or vice versa.

In our example, the solution \(x \leq 50\) was expressed in interval notation as \((-\infty, 50]\), indicating that \(x\) can be any real number up to and including 50.

Distributive Property

The distributive property is a fundamental algebraic principle that involves multiplying a single term by each term inside parentheses. This property can be stated as: \(a(b + c) = ab + ac\).
In our example, we applied the distributive property to \(0.08(250 - x)\), resulting in \(0.08 \times 250 - 0.08 \times x\), which simplifies to \(20 - 0.08x\).
Using the distributive property helps us eliminate parentheses and rewrite expressions in a simpler form. It is very useful when simplifying equations or inequalities because it enables us to combine and rearrange terms effectively, which is often a necessary step in solving mathematical equations.

Combining Like Terms

Combining like terms involves simplifying expressions by gathering similar terms. This step streamlines the solution process by reducing the number of terms in an equation or inequality.
In our inequality \(0.06x + 20 - 0.08x \geq 19\), the terms \(0.06x\) and \(-0.08x\) are like terms since they both contain the variable \(x\). Combining them simplifies the expression:

• \(0.06x - 0.08x = -0.02x\)

Combining like terms is crucial for organizing equations into a manageable form, allowing us to see more clearly what steps are necessary next. This simplification is a step toward isolating variables and finding solutions in mathematical problems.