Problem 31 - Intermediate Algebra

Question

Solve each compound inequality, if possible. Graph the solution set (if one exists) and write it using interval notation. \(x+2<-\frac{1}{3} x\) and \(-6 x<9 x\)

Step-by-Step Solution

Answer

The solution set is empty: \(x \text{ such that } x < -\frac{3}{2} \cap x > 0 = \emptyset \).

Step 1: Understanding the Compound Inequality

The given problem is a compound inequality consisting of two separate inequalities connected by 'and'. These inequalities are: \(x + 2 < -\frac{1}{3}x\) and \(-6x < 9x\). We need to solve each inequality separately, and then find the common solution set (intersection) for both inequalities.

Step 2: Solve the First Inequality

Rearrange the inequality \(x + 2 < -\frac{1}{3}x\) by adding \(\frac{1}{3}x\) to both sides to eliminate \(\frac{1}{3}x\) from the right side: \(x + \frac{1}{3}x < -2\). Combine like terms to get \(\frac{4}{3}x < -2\). Multiply both sides by \(\frac{3}{4}\) to isolate \(x\): \(x < -\frac{3}{2}\).

Step 3: Solve the Second Inequality

Rearrange the inequality \(-6x < 9x\) by adding \(6x\) to both sides: \(0 < 15x\). To solve for \(x\), divide both sides by 15: \(0 < x\) or \(x > 0\).

Step 4: Find the Intersection of the Solutions

The solution to the compound inequality requires both conditions to be true simultaneously. Thus, we need the intersection of the solutions \(x < -\frac{3}{2}\) and \(x > 0\). Since these intervals do not overlap, there is no common solution.

Step 5: Graph the Solution Set

Since there is no overlap in the solutions \(x < -\frac{3}{2}\) and \(x > 0\), there is no graph to represent a solution set for the compound inequality. The solution is an empty set.

Step 6: Write the Solution in Interval Notation

Reflecting on the intersection of intervals \((-\infty, -\frac{3}{2})\) and \((0, \infty)\), we conclude that there is no common interval that satisfies both conditions. Therefore, the solution in interval notation is \(\emptyset\).

Key Concepts

Inequality Solutions

Solving inequalities involves finding values for the variable that make the inequality true. For compound inequalities, there are usually two or more inequalities joined by 'and' or 'or'. In our problem, we have the two inequalities linked by 'and':

• \(x + 2 < -\frac{1}{3}x\)
• \(-6x < 9x\)

To solve them, tackle each one separately:

• move terms around until the variable is isolated on one side.
• perform operations such as addition, subtraction, multiplication, or division across all terms carefully, particularly when multiplying or dividing by a negative number, as this flips the inequality sign.

Once each inequality is solved, determine the values that satisfy both if linked by 'and' or satisfy either if linked by 'or'. The solution is the set of these values.

Interval Notation

Interval notation is a concise way to express a range of values along a number line.
In our exercise, we solved for conditions:

• \(x < -\frac{3}{2}\)
• \(x > 0\)

However, for a compound inequality connected by 'and', we need our solutions to overlap. Here’s how they look in interval notation:

• \(x < -\frac{3}{2}\) is written as \((-\infty, -\frac{3}{2})\)
• \(x > 0\) is \((0, \infty)\)

Since there is no overlapping region for our intervals, our final solution in interval notation is \(\emptyset\), meaning there are no values that satisfy both conditions simultaneously.

Graphing Inequalities

Graphing inequalities on a number line provides a visual representation of solution sets:
Draw a number line:

• For \(x < -\frac{3}{2}\), shade the region to the left of \(-\frac{3}{2}\) with an open circle at \(-\frac{3}{2}\).
• For \(x > 0\), shade to the right of 0 with an open circle at 0.

The intersection (or shared region) between the shaded parts represents the solution to the compound inequality. If shading doesn't overlap, as in our case, it indicates that no shared solution exists, thus the solution is an empty set.
Graphing these inequalities helps intuitively understand why no solution exists when intervals do not intersect.