Problem 58 - Intermediate Algebra

Question

Solve each inequality. Graph the solution set and write it using interval notation. $$ 0.05+0.8 x \leq 0.5 x-0.7 $$

Step-by-Step Solution

Answer

The solution is \(x \leq -2.5\), or interval notation: \((-\infty, -2.5]\).

Step 1: Move Variables to One Side

Start by moving all terms involving the variable \(x\) to one side of the inequality. To do this, subtract \(0.5x\) from both sides: \[0.05 + 0.8x - 0.5x \leq -0.7\]Simplifying this step gives:\[0.05 + 0.3x \leq -0.7\]

Step 2: Isolate the Variable

Next, isolate the term with the variable \(x\) by subtracting \(0.05\) from both sides:\[0.3x \leq -0.7 - 0.05\]This simplifies to:\[0.3x \leq -0.75\]

Step 3: Solve for x

Divide both sides of the inequality by \(0.3\) to solve for \(x\):\[x \leq \frac{-0.75}{0.3}\]Calculating this gives:\[x \leq -2.5\]

Step 4: Graph the Solution Set

On a number line, draw a solid circle at \(-2.5\) to represent that it includes this value (as the inequality uses \(\leq\)) and shade to the left, indicating all values less than \(-2.5\).

Step 5: Express in Interval Notation

Since \(x\) takes all values less than or equal to \(-2.5\), the solution in interval notation is:\[(-\infty, -2.5]\]

Key Concepts

Graphing Inequalities

Graphing inequalities is a crucial skill in understanding how to visually represent the solutions of inequalities on a number line. In the inequality \( x \leq -2.5 \), our goal is to display all possible values of \( x \) that satisfy this condition. To graph this:

• First, locate the number \(-2.5\) on the number line. This is a key point because it marks the boundary of our solution set.
• Draw a solid circle around \(-2.5\). A solid circle means that \(-2.5\) is included in the set of solutions, as the inequality symbol used is \( \leq \).
• From \(-2.5\), shade the entire portion of the number line to the left. This shading indicates all numbers less than \(-2.5\) are also part of the solution.

By graphing the inequality, you can easily see which numbers satisfy the condition. It's a simple, yet effective method to visualize solutions.

Interval Notation

Interval notation offers a concise way to describe a range of numbers that are solutions to an inequality. For the inequality \( x \leq -2.5 \), we express the solution in interval notation as \((-\infty, -2.5]\). Here's how interval notation works:

• The round bracket \((-\infty\)) denotes that infinity is not a specific number but an idea, and thus cannot be included; it is always represented with a round bracket.
• The square bracket \([-2.5]\) indicates that the value \(-2.5\) is included in the interval. This matches the inclusive nature of the "less than or equal to" inequality.
• The comma between \(-\infty\) and \(-2.5\) clearly separates the starting point from the end point of the interval.

Interval notation is not only a shorthand method but also clarifies whether boundaries are included or excluded in the solution.

Solving Linear Inequalities

Solving linear inequalities involves finding all the possible values of the variable that make the inequality true. Taking \(0.05 + 0.8x \leq 0.5x - 0.7\) as an example, follow these steps to solve it:

• Firstly, group like terms by moving all terms containing \(x\) to one side. Subtract \(0.5x\) from both sides to get \(0.05 + 0.3x \leq -0.7\).
• Next, isolate \(x\) by removing constant terms from its side. Subtract \(0.05\) from both sides, leading to \(0.3x \leq -0.75\).
• To solve the inequality for \(x\), divide each side by \(0.3\) to get \(x \leq -2.5\).

The main principle of solving linear inequalities is similar to solving equations. However, remember to reverse the inequality sign if you multiply or divide by a negative number, though that step is not applicable in this example. Solving inequalities carefully following these rules ensures accurate solutions.