Problem 57 - Intermediate Algebra

Question

Solve each inequality. Graph the solution set and write it using interval notation. $$ 0.4 x+0.4 \leq 0.1 x+0.85 $$

Step-by-Step Solution

Answer

The solution is \((-\infty, 1.5]\).

Step 1: Set Up the Inequality

We start with the given inequality: \( 0.4x + 0.4 \leq 0.1x + 0.85 \). Our goal is to find the values of \( x \) that satisfy this inequality.

Step 2: Organize the Terms

Subtract \( 0.1x \) from both sides to move \( x \)-terms together: \( 0.4x - 0.1x + 0.4 \leq 0.85 \). This simplifies to \( 0.3x + 0.4 \leq 0.85 \).

Step 3: Eliminate the Constant

Subtract 0.4 from both sides to isolate the \( x \)-term: \( 0.3x + 0.4 - 0.4 \leq 0.85 - 0.4 \). This simplifies to \( 0.3x \leq 0.45 \).

Step 4: Solve for x

Divide both sides by 0.3 to solve for \( x \): \( x \leq \frac{0.45}{0.3} \). This simplifies to \( x \leq 1.5 \).

Step 5: Write in Interval Notation

Express the solution \( x \leq 1.5 \) in interval notation. The solution set is \( (-\infty, 1.5] \).

Step 6: Graph the Solution

On a number line, draw a solid dot at 1.5 to indicate that it's included in the solution. Shade the line to the left of 1.5, indicating all numbers in that direction are solutions.

Key Concepts

Solving Inequalities

To solve an inequality involves finding all possible values of the variable that make the inequality true. In our exercise, we started with the inequality \( 0.4x + 0.4 \leq 0.1x + 0.85 \) and aimed to identify all values of \( x \) that satisfy this condition. Here's a step-by-step guide to get you through it:

• Start with the original inequality and aim to isolate the variable \( x \) on one side.
• Use basic algebra to move all terms containing \( x \) to one side of the inequality and constants to the other.
• Simplify the expression by combining like terms or by performing arithmetic operations such as addition, subtraction, multiplication, or division.
• Always remember, when you multiply or divide an inequality by a negative number, the direction of the inequality reverses.

For our case, after manipulating the inequality, we determined that \( x \leq 1.5 \). Each of these steps helps narrow down which values \( x \) can assume to keep our original statement valid.

Interval Notation

Interval notation is a method for expressing sets of solutions. It's particularly useful for inequalities because it succinctly represents a continuous range of values.Here's how to interpret it:

• An interval \((-\infty, a] \) means that the solution includes all numbers less than or equal to \( a \).
• If the interval were \((a, \infty) \), it would mean all numbers greater than \( a \).
• Parentheses \(()\) show that an endpoint is not included, whereas square brackets \([]\) indicate inclusion of the endpoint.

In our scenario, \( x \leq 1.5 \) is written as \((-\infty, 1.5] \), meaning all real numbers up to and including 1.5 satisfy the inequality. It's a clear and concise way of describing solution sets and is commonly used in mathematics.

Graphing Inequalities

Graphing inequalities provides a visual representation of the solutions, helping you easily identify the range of values that satisfy the inequality.In this exercise:

• You begin by drawing a number line that includes the point 1.5.
• Next, represent the inequality \( x \leq 1.5 \) by placing a solid dot at 1.5 to show that 1.5 itself is a part of the solution.
• Shade the number line to the left of 1.5, indicating that all values less than or equal to 1.5 are part of the solution.

This shading technique helps visualize where the solution set lies relative to the points drawn on the number line. Graphing is a powerful tool to understand inequalities, and it's especially helpful for more complicated compound inequalities.

Algebraic Manipulation

Algebraic manipulation involves rearranging, adding, or subtracting expressions to simplify or solve equations and inequalities. It often includes strategic moves to isolate the variable of interest.For our exercise, here's what we did:

• Subtracted \( 0.1x \) from both sides to bring all \( x \)-terms to one side, simplifying \( 0.4x - 0.1x \) to \( 0.3x \).
• Moved constant terms, in this case by subtracting \( 0.4 \) from both sides, leaving us with \( 0.3x \leq 0.45 \).
• Finally, divided by 0.3 to solve for \( x \), resulting in \( x \leq 1.5 \).

Remember that each algebraic step should maintain the equivalence or inequality of the original expression. Clear and methodical manipulation leads to finding the correct solution to inequalities and is crucial for problem-solving in mathematical contexts.