Problem 56 - Elementary and Intermediate Algebra

Question

Solve each compound inequality, if possible. Graph the solution set (if one exists) and write it using interval notation. $$ -\frac{x}{4}>-2.5 \text { and } 9 x>2(4 x+5) $$

Step-by-Step Solution

Answer

No solution; the solution set is empty (\(\emptyset\)).

Step 1: Solve the first inequality

The first inequality is \(-\frac{x}{4} > -2.5\). Multiply both sides by \(-4\) to solve for \(x\), remembering to reverse the inequality sign when multiplying or dividing by a negative number:\[x < 10\]

Step 2: Solve the second inequality

The second inequality is \(9x > 2(4x + 5)\). First, distribute the \(2\) on the right-hand side:\[9x > 8x + 10\]Next, subtract \(8x\) from both sides to isolate \(x\):\[x > 10\]

Step 3: Find the intersection of the solution sets

The solutions to the individual inequalities are \(x < 10\) and \(x > 10\). The intersection of these two inequalities is an empty set because there is no \(x\) that satisfies both \(x < 10\) and \(x > 10\).

Step 4: Represent the solution set

Since the solution set is empty, there is no graph, and the interval notation for an empty set is written as \( \emptyset \).

Key Concepts

inequality solving

Solving inequalities is similar to solving equations, yet there are crucial differences. When we solve inequalities, our goal is to determine the range of values that satisfy the inequality. In the exercise provided, we deal with two inequalities connected by an "and" condition, meaning a number must satisfy both to be a part of the solution set.

• The first inequality, \(-\frac{x}{4} > -2.5\), requires solving by isolating \(x\). By multiplying both sides of the equation by \(-4\) (and reversing the inequality sign due to multiplication by a negative), we find that \(x < 10\).
• The second inequality, \(9x > 2(4x + 5)\), requires distribution and simplification. By simplifying, you first find it becomes \(9x > 8x + 10\); after subtracting \(8x\) from both sides, you get \(x > 10\).

Inequality solving often involves shifting terms, reversing inequality signs, and logical reasoning to find feasible solutions. Remember, though inequalities resemble equations, the operations could lead to multiple or no solutions at all.

interval notation

After solving the inequalities, we represent solutions concisely using interval notation. This form simplifies expressing ranges or specific values succinctly

• Open intervals, such as \((a, b)\), indicate that \(a\) and \(b\) are not part of the solution, whereas closed intervals, using brackets \([a, b]\), include both boundary points.
• When an inequality has no valid solutions, as in our exercise, it's expressed as an empty set in interval notation, written as \(\emptyset\).

Interval notation beautifully condenses solution sets, making them easy to interpret and use in further mathematical operations.

solution set visualization

Visualizing the solution set of an inequality on a number line assists in understanding the range of possible solutions. Graphing becomes particularly helpful with compound inequalities where multiple constraints apply.

• For \(x < 10\) and \(x > 10\) simultaneously, no points exist that satisfy both conditions. Thus, the solution set is empty, making graphing impractical in the usual sense.
• When graphing is possible, use open circles for non-inclusive boundaries and filled circles for inclusive boundaries. Arrows indicate the direction of the inequality.

By visualizing inequalities, you reinforce understanding of how solutions are distributed along the number line, highlighting overlaps and gaps distinctly. A clear graph serves as a checkpoint, verifying the algebraic solution.