Problem 62 - Introductory Algebra for College Students

Question

Use both the addition and multiplication properties of inequality to solve each inequality and graph the solution set on a number line. \(5-3 x \geq 20\)

Step-by-Step Solution

Answer

The solution to the inequality \(5 - 3x \geq 20\) is \(x \leq -5\). On a number line, the point at -5 is marked and the line extends to the left.

Step 1: Subtract 5 on Both Sides

In order to isolate the term with the variable \(x\), subtract 5 from both sides of the inequality: \(5 - 3x - 5 \geq 20 - 5\), which simplifies to \(-3x \geq 15\).

Step 2: Divide Both Sides by -3

Now, to solve for \(x\), we have to divide both sides of the inequality by -3. Take note that when we divide or multiply by a negative number, the direction of inequality changes: \(-3x / -3 \leq 15 / -3\), leading to \(x \leq -5\).

Step 3: Graph the Solution on a Number Line

Draw a number line, and mark -5 on it. Since the solution to the inequality is \(x \leq -5\), the arrow should point to the left. Put a closed circle on -5 to include the number -5 in the solution set.

Key Concepts

Solving Inequalities

When solving inequalities, the main goal is to find the set of values for the variable that make the inequality true. Like solving regular equations, we aim to isolate the variable on one side. However, inequalities involve some additional considerations, particularly when using multiplication or division by negative numbers. This may involve several steps:

• Eliminating any constants from both sides of the inequality using addition or subtraction.
• Dividing or multiplying both sides to solve for the variable, ensuring to switch the inequality sign if multiplying or dividing by a negative number.
• Checking your solution by plugging some values from the solution set back into the original inequality to confirm they satisfy the condition.

Remember, solving inequalities is like solving equations, but we have to be careful about the inequality direction.

Graphing Inequalities

Graphing inequalities involves representing the solution sets visually on a number line. This helps to easily identify which values satisfy the inequality. For example:

• Identify the endpoint from the inequality (e.g., in the solution to \(x \leq -5\), the endpoint is -5).
• Use a closed circle if the inequality is \(\geq\) or \(\leq\), showing that the endpoint is included in the solution set.
• Draw an arrow from the endpoint in the direction of the inequality. For \(x \leq -5\), the arrow points to the left.

Graphing offers a straightforward way to visualize the range of values fulfilling the inequality.

Addition Property of Inequality

The addition property of inequality states that you can add the same number to both sides of an inequality without changing its direction. This property is handy when isolating variables. For instance, in solving the inequality \(5 - 3x \geq 20\), we subtracted 5 from both sides to maintain balance.

• This simplification helps to manage the inequality and bring us closer to isolating the variable.
• Always ensure consistency by performing the operation on both sides to preserve the inequality's truth.

Remember, the addition property keeps the inequality intact, so the overall balance stays the same.

Multiplication Property of Inequality

The multiplication property of inequality highlights a crucial rule: when you multiply or divide by a negative number, the inequality sign flips. This was evident in the solution where we divided both sides of \(-3x \geq 15\) by -3, resulting in the sign flipping to \(x \leq -5\).

• Always pay attention when multiplying or dividing by a negative number, as it changes the inequality's direction.
• This is due to reversing the overall order when negatives are introduced.
• Double-check your results by ensuring that the revised inequality statement correctly represents the solution set.

This rule is essential to accurately solving inequalities involving negative coefficients.