Problem 20 - Algebra for College Students

Question

Solve each inequality. Graph the solution set, and write it using interval notation. \(-\frac{2}{3} x \leq 12\)

Step-by-Step Solution

Answer

x \geq -18; [-18, \infty)

Step 1 - Isolate the variable

To solve the inequality \(-\frac{2}{3} x \leq 12\), first isolate the variable x. Start by multiplying both sides of the inequality by the reciprocal of -\(\frac{2}{3}\). Since we multiply or divide by a negative number, we must reverse the inequality sign. The reciprocal of -\(\frac{2}{3}\) is -\(\frac{3}{2}\): \[ \left( -\frac{3}{2} \right) \left( -\frac{2}{3} x \right) \geq 12 \left( \frac{3}{2} \right) \]

Step 2 - Simplify

Simplify both sides of the inequality: \[ x \geq 12 \times -\frac{3}{2} \] \[ x \geq -18 \]

Step 3 - Graph the solution set

On a number line, shade all numbers greater than or equal to -18. This means starting from -18 and shading to the right with a closed circle at -18.

Step 4 - Write the solution in interval notation

The solution in interval notation is written as: \([-18, \infty)\). This interval includes all numbers from -18 to positive infinity, and it includes -18.

Key Concepts

Inequalities

Inequalities are mathematical statements that show the relationship between two expressions that are not necessarily equal. In the given exercise, we have an inequality: equation: \[ -\frac{2}{3} x \leq 12 \]
To solve inequalities, we use similar techniques as we do for solving equations, with one key difference: whenever we multiply or divide both sides of an inequality by a negative number, we must reverse the inequality sign. Inequalities can be of different types, such as <, >, ≤, and ≥.
Understanding inequalities is crucial because they allow us to find ranges of values rather than a single specific value, giving us more flexibility and real-world applicability.

Interval Notation

Once we solve an inequality, we often express the solution using interval notation. Interval notation is a way of writing subsets of the real number line. For the given solution, we obtained:\[ x \geq -18 \]
Interval notation for this inequality is:” \[ [-18, \infty) \]
Here,

• '[' means the interval includes -18.
• ',' denotes a separation between the two bounds.
• ‘∞’ means it extends to positive infinity.
• ')' indicates that the interval does not include ∞.

Interval notation is beneficial because it provides a concise way to express a range of numbers.

Reciprocal

In this exercise, we used the reciprocal of a fraction to isolate the variable. The reciprocal of a number is what you multiply that number by to get 1. For the fraction \[ -\frac{2}{3} \], we need to identify its reciprocal. We flip the numerator and the denominator, so the reciprocal is\[ -\frac{3}{2} \].
When we multiply both sides of our inequality \[ -\frac{2}{3} x \leq 12 \] by the reciprocal \[ -\frac{3}{2} \], we must remember to reverse the inequality sign. This step allows us to isolate the variable x and simplify to find the solution.

Graphing Inequalities

To visually represent the solution, we graph it on a number line. For the inequality \[ x \geq -18, \] we need to shade all values greater than or equal to -18. Here's how to graph it:

• Draw a number line.
• Locate -18 on the number line.
• Use a closed circle (or dot) at -18 to indicate that -18 is included in the solution.
• Shade the number line to the right of -18, extending to positive infinity.

This shaded portion represents all numbers that satisfy the inequality, creating a clear visual for understanding the solution set.