Problem 8 - Elementary Algebra

Question

Graph the equations and inequalities. $$ -16 \leq 5 x-1 \leq-11 $$

Step-by-Step Solution

Answer

Answer: The range of x values that satisfy the compound inequality is $$-3 \leq x \leq -2$$.

Step 1: Separate the inequalities

We have a compound inequality, so we need to separate it into two separate inequalities: $$ -16 \leq 5x - 1 \\ 5x - 1\leq -11 $$

Step 2: Solve the first inequality

To solve the inequality $$-16 \leq 5x - 1$$, we'll need to isolate x: 1. Add 1 to both sides: $$-15 \leq 5x$$ 2. Divide both sides by 5: $$-3\leq x$$

Step 3: Solve the second inequality

To solve the inequality $$5x - 1\leq -11$$, we'll need to isolate x: 1. Add 1 to both sides: $$5x\leq -10$$ 2. Divide both sides by 5: $$x\leq -2$$

Step 4: Combine the inequalities

Now we'll put the two inequalities together to find the range of x that satisfies both inequalities: $$ -3\leq x\leq -2 $$

Step 5: Graph the solution

To graph this inequality, create a number line with x values from -3 to -2. Make sure to include closed dots at -3 and -2 to indicate that these values are included in the solution. Shade the region between the two dots to represent the range of x values that satisfy the inequality: $$ -3 \leq x \leq -2 $$

Key Concepts

Compound Inequalities

Compound inequalities are mathematical expressions that consist of two or more simple inequalities connected by the words "and" or "or." The goal is to find all possible solutions that satisfy both (for "and") or either (for "or") inequalities.
In this exercise, we have a compound inequality that uses "and" because it combines two inequalities:

• \(-16 \leq 5x - 1\)
• \(5x - 1 \leq -11\)

Such an inequality requires solutions to satisfy both conditions simultaneously. We solve compound inequalities by breaking them into individual inequalities, solving each one, and then considering the intersection of these solutions.

Solving Inequalities

Solving inequalities follows a similar process as solving equations, but with a few additional considerations. Primarily, when multiplying or dividing both sides of an inequality by a negative number, the direction of the inequality sign must be switched.
For the given compound inequality:

• First, separate into \(-16 \leq 5x - 1\) and \(5x - 1 \leq -11\).


• Solve the first inequality: Add 1 to both sides, then divide by 5, resulting in \(-3 \leq x\).


• Solve the second inequality: Add 1 to both sides, then divide by 5, resulting in \(x \leq -2\).

By finding the value of \(x\) that satisfies both inequalities, you form the complete solution. Here, the solution \(-3 \leq x \leq -2\) reflects the intersection of the solutions where both conditions are met.

Graphing Inequalities

Graphing inequalities visually represents the solution set on a number line or coordinate plane. It helps to easily identify all possible solutions in a given range.
In the compound inequality \(-3 \leq x \leq -2\), graphing translates solution values into visual information:
Place closed dots at points -3 and -2, showing that these endpoints are included in the solution set. This inclusion is indicated by the 'less than or equal to' symbol (\(\leq\)).
Once these points are marked, shade the region between the closed dots to show all possible values of \(x\) that satisfy both parts of the compound inequality.

Number Line Representation

A number line representation is a powerful visual tool for understanding and communicating the solution to inequalities. It provides clarity and helps in quickly determining which values are included within an inequality's solution set.
In our example, the number line for \(-3 \leq x \leq -2\) includes specific elements:

• A horizontal line (number line) with intervals marked.


• Closed dots at -3 and -2, indicating that these boundary numbers are inclusive in the solution set.


• A shaded segment between these points, representing all real numbers that satisfy the inequality.

This method makes complex math concepts more accessible, as it translates algebraic solutions into an easily interpretable visual form.