Problem 13 - Intermediate Algebra

Question

For Problems \(1-18\), 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 in interval notation is \((-\infty, 50]\).

Step 1: Distribute the constant in the inequality

Expand the term \(0.08(250-x)\) by multiplying each part inside the parenthesis by 0.08.\[0.06x + 0.08 \times 250 - 0.08x \geq 19\]This simplifies to:\[0.06x + 20 - 0.08x \geq 19\]

Step 2: Combine like terms

Combine the \(x\) terms on the left side:\[(0.06x - 0.08x) + 20 \geq 19\]This simplifies to:\[-0.02x + 20 \geq 19\]

Step 3: Isolate the variable term

Subtract 20 from both sides to start getting \(x\) by itself.\[-0.02x + 20 - 20 \geq 19 - 20\]This results in:\[-0.02x \geq -1\]

Step 4: Solve for \(x\)

Divide each side by \(-0.02\) to solve for \(x\). Don't forget to flip the inequality sign because you are dividing by a negative number.\[x \leq \frac{-1}{-0.02}\]Which simplifies to:\[x \leq 50\]

Step 5: Express the solution in interval notation

Since \(x\) is less than or equal to 50, the solution in interval notation is:\[(-\infty, 50]\]

Key Concepts

Interval Notation

Interval notation is a way of writing the set of solutions to an inequality, making it easy to see the range of values that satisfy the condition. Instead of listing all possible numbers, interval notation uses brackets to show where the solutions start and end.

• "(" or ")" indicates that the endpoint is not included (open interval).
• "[" or "]" indicates that the endpoint is included (closed interval).

In our solution, the inequality implies that all values of \(x\) are less than or equal to 50. This is captured as \((-\infty, 50]\).

The open interval \((-\infty, 50]\) starts from negative infinity and includes all numbers up to (and including) 50. The infinity symbol \(-\infty\) always accompanies an open bracket because infinity is a concept, not a specific number.

Distribution in Algebra

Distribution is a key algebraic property used to simplify expressions or inequations, especially when dealing with parentheses. It follows the distributive property formula: \[a(b + c) = ab + ac\]

This operation allows you to multiply each term inside the parenthesis by a factor outside it.

In the given inequality: \[0.08(250 - x)\] we distribute \(0.08\) to both \(250\) and \(-x\). This expands to: \[0.08 \times 250 - 0.08 \times x\]

After distribution, you simplify each part to manage and combine terms more efficiently. Distribution is a stepping stone in solving many algebraic problems and understanding where it applies helps maintain clarity when managing expressions.

Combining Like Terms

Combining like terms is a method used to simplify expressions or inequations. Like terms are terms that have the same variables raised to the same power. These can be combined by adding or subtracting their coefficients.

• Consider the expression: \(0.06x + 20 - 0.08x\)

In this case, \(0.06x\) and \(-0.08x\) are like terms because they both involve \(x\). When we combine them, we calculate: \[0.06x - 0.08x = -0.02x\]

By grouping these like terms together, you create a simpler form: \(-0.02x + 20\).

This simplification makes it easier to solve the inequality by isolating variables on one side and constants on the other. Mastering the skill of combining like terms can significantly streamline your problem-solving process.