Problem 15 - Intermediate Algebra

Question

For Problems \(1-18\), solve each of the inequalities and express the solution sets in interval notation. $$ 0.09 x+0.1(x+200)>77 $$

Step-by-Step Solution

Answer

The solution is \((300, \infty)\).

Step 1: Expand the Expression

The inequality given is \(0.09x + 0.1(x + 200) > 77\). Start by expanding the term \(0.1(x + 200)\). This will give us \(0.1x + 20\).

Step 2: Combine Like Terms

Now combine the like terms in the inequality: \(0.09x + 0.1x + 20 > 77\). This simplifies to \(0.19x + 20 > 77\).

Step 3: Isolate the Variable Term

Subtract 20 from both sides of the inequality to isolate the term with \(x\): \(0.19x > 77 - 20\). This simplifies to \(0.19x > 57\).

Step 4: Solve for the Variable

Divide both sides by 0.19 to solve for \(x\): \(x > \frac{57}{0.19}\). Calculating this gives \(x > 300\).

Step 5: Write the Solution in Interval Notation

Express the solution \(x > 300\) in interval notation. The solution in interval notation is \((300, \infty)\).

Key Concepts

Interval Notation

Interval notation is a way of writing subsets of the real number line. It's often used to express solutions to inequalities. This method simplifies the expression of range, using brackets and parentheses to denote inclusive or exclusive bounds.
For example, when expressing solutions for inequalities:

• An open parenthesis "](" or ")" is used when the number is not included in the solution set (exclusive).
• A closed bracket "[" or "]" indicates the number is included (inclusive).

When you have an inequality like \(x > 300\), you express this in interval notation as \((300, \infty)\). The parenthesis next to 300 tells us that 300 is not part of the solution, and the infinity symbol signifies that there is no upper bound.

Expanding Expressions

To solve mathematical expressions efficiently, it's vital to properly expand them. Expanding an expression usually involves distributing a multiplier over terms inside parentheses.
In the inequality \(0.09x + 0.1(x + 200) > 77\), you see multiplication with 0.1 across terms inside the parentheses. This results in:

• \(0.1 \times x\) becoming \(0.1x\)
• \(0.1 \times 200\) becoming 20

This step is crucial as it simplifies the terms making it easier to combine and further solve the expression.

Combining Like Terms

The next logical step after expanding expressions is to combine like terms. Like terms have the same variable raised to the same power. This consolidation is crucial since it simplifies expressions further.
In the example \(0.09x + 0.1x + 20 > 77\), the terms \(0.09x\) and \(0.1x\) are like terms because they both have the variable \(x\). Combining these yields:

• \(0.09x + 0.1x = 0.19x\)

This simplification transforms the expression into \(0.19x + 20 > 77\), making it easier to isolate the variable.

Solving Inequalities

Solving inequalities involves isolating the variable so that the inequality can be rewritten in simple terms. Once we have simplified an inequality, the goal is to have the variable by itself on one side.From the step \(0.19x + 20 > 77\):

• Subtract 20 from both sides: \(0.19x > 57\).
• Now, divide each side by 0.19 to isolate \(x\): \(x > \frac{57}{0.19}\).
• This calculation gives \(x > 300\).

The inequality is now solved, expressing \(x\) in simpler terms. The next and final step is to present this as an interval, like \((300, \infty)\), to effectively communicate the solution of the inequality.