Problem 63 - Intermediate Algebra

Question

Solve each compound inequality, if possible. Graph the solution set (if one exists) and write it using interval notation. \(\frac{x}{0.7}+5>4\) and \(-4.8 \leq \frac{3 x}{-0.125}\)

Step-by-Step Solution

Answer

The solution is \((-0.7, 0.2]\).

Step 1: Solve the First Inequality

The first inequality is \( \frac{x}{0.7} + 5 > 4 \). Begin by isolating the fraction term by subtracting 5 from both sides: \( \frac{x}{0.7} > -1 \). Then, multiply both sides by 0.7 to solve for \( x \): \( x > -0.7 \).

Step 2: Solve the Second Inequality

The second inequality is \( -4.8 \leq \frac{3x}{-0.125} \). First, multiply both sides by \(-0.125\) to eliminate the fraction: \( -4.8 \times -0.125 \geq 3x \), which simplifies to \( 0.6 \geq 3x \). Divide both sides by 3 to solve for \( x \): \( 0.2 \geq x \) or \( x \leq 0.2 \).

Step 3: Find the Common Solution

The solution to the compound inequality requires finding values of \( x \) that satisfy both \( x > -0.7 \) and \( x \leq 0.2 \). The intersection (common solution) is \( -0.7 < x \leq 0.2 \).

Step 4: Express in Interval Notation

The solution can be expressed in interval notation as \((-0.7, 0.2]\).

Step 5: Graph the Solution Set

On a number line, plot an open circle at \(-0.7\) (indicating \( x > -0.7 \)) and a closed circle at \(0.2\) (indicating \( x \leq 0.2 \)). Shade the region between these two points to represent the solution set \((-0.7, 0.2]\).

Key Concepts

Interval Notation

Interval notation is a convenient way of expressing a range of values for which a variable, such as \( x \), satisfies certain conditions. Instead of listing or describing the values one by one, interval notation offers a shorthand.

For the compound inequality in the exercise, we found that the solution is that \( x \) must be greater than \( -0.7 \) and less than or equal to \( 0.2 \). In interval notation, we write this as \((-0.7, 0.2]\).

Let's break this down:

• The round parenthesis \( ( \) at the \( -0.7 \) indicates that \( -0.7 \) itself is not included in the solution (i.e., \( x \) must be greater than \( -0.7 \)).
• The square bracket \( ] \) at \( 0.2 \) signifies that \( 0.2 \) is included (i.e., \( x \) can equal \( 0.2 \)).

Interval notation is especially helpful in clearly and concisely conveying the solution set, especially when combined with graphing.

Solution Set

The solution set is the collection of all values that satisfy a specific inequality or a system of inequalities. It's like the group of answers that work for our problem.

In this exercise, we dealt with a compound inequality involving two separate inequalities. Our task was to find the set of \( x \) values that meet both criteria.
- From the first inequality, \( x > -0.7 \).- From the second inequality, \( x \leq 0.2 \).

The solution set is the overlap of these conditions, meaning values that fulfill both at the same time. For \( x \) to satisfy both, it must be greater than \( -0.7 \) but also \( x \leq 0.2 \). Hence, values from just a bit over \( -0.7 \) up to and including \( 0.2 \) make up our solution set.

This dual satisfaction is key in compound inequalities, as they often require understanding which parts of each individual inequality combine to form the complete solution.

Graphing Inequalities

Graphing inequalities helps visualize the solution set by depicting which values of \( x \) are included.

In our exercise, we graph the compound inequality solution set \((-0.7, 0.2]\) on a number line. Here's how it works:

• Draw a number line that includes the points \( -0.7 \) and \( 0.2 \).
• At \( -0.7 \), place an open circle, which shows that this point is not part of the solution (\( x > -0.7 \)).
• At \( 0.2 \), place a closed circle to indicate this point is part of the solution (\( x \leq 0.2 \)).
• Shade the area between these two points to show all values \( x \) that satisfy the inequality.

Graphing is a powerful tool as it provides a visual representation, making it easier to understand and communicate the possible solutions. Plus, it quickly shows both endpoints and if they are included or excluded in the solution.