Problem 9 - Prealgebra

Question

Solve each inequality. Check your solution. Then graph the solution on a number line. $$-8 z \leq-24$$

Step-by-Step Solution

Answer

The solution is \( z \geq 3 \). Graph: dot at 3, arrow right.

Step 1: Isolate the Variable

To isolate the variable \( z \), first divide both sides of the inequality \( -8z \leq -24 \) by \( -8 \). When you divide or multiply an inequality by a negative number, the inequality sign flips direction. Thus, we will have \( z \geq 3 \).

Step 2: Verify the Solution

Substitute \( z = 3 \) back into the original inequality to check the solution: \(-8(3) = -24\) satisfies the condition \(-8z \leq -24\). Check \( z > 3 \) as well to ensure all numbers greater than or equal to 3 also satisfy the inequality. For instance, for \( z = 4 \), \(-8(4) = -32\), which is indeed less than \(-24\).

Step 3: Graph the Solution

On a number line, draw a solid dot on the number 3 to indicate that 3 is included in the solution. Then draw a line extending to the right from 3 to show all numbers greater than 3 are part of the solution.

Key Concepts

Understanding How to Check Solutions for Inequalities

When solving inequalities, it is vital to verify that your solution is correct. This involves substituting the solution back into the original inequality.

• Begin by taking the result of your work, which in this case is the variable inequality solution, like our example where we found \(z \geq 3\).
• Substitute the boundary value, \(z = 3\), into the original inequality: \(-8z \leq -24\).
• Calculate to see if it holds true. For \(z = 3\), you get \(-8(3) = -24\), which is equal and thus satisfies the inequality condition.

It's also good practice to test the inequality with values greater than the boundary. Try substituting \(z = 4\) to check: \(-8(4) = -32\), which is \(-32 < -24\). This confirms that all numbers greater than or equal to 3 satisfy the inequality. By checking these values, you ensure that your solution encompasses the full range of possibilities indicated by the inequality.

Graphing Inequalities Made Simple

Graphing inequalities involves a constant process of problem-solving and visualization,To represent an inequality solution on a number line effectively:

• Identify the boundary point of the solution. In our example, it's the number 3.
• Determine if the boundary is inclusive or exclusive. An inclusive boundary uses a solid dot because the number is part of the solution (\(z \geq 3\) requires a solid dot on 3).
• Extend a line to the right if the inequality symbol indicates greater than or equal to, representing all numbers greater than the boundary.

Through graphing, inequalities are transformed from abstract concepts into a visual representation. This aids in understanding where solutions lie on the number line, making the information more concrete and accessible.

Using Number Line Representation for Inequalities

A number line is a useful tool for visualizing solutions to inequalities. It provides a clear picture of which numbers are included in the solution set.

Steps to Use a Number Line:

• Identify the critical value where the inequality begins, such as 3 in our inequality \(z \geq 3\).
• Place a solid dot on this value if it is included in the solution.
• Draw an arrow from the dot in the direction that includes all the numbers satisfying the inequality. For \(z \geq 3\), the arrow points right.

By visually mapping the solutions, we can quickly see and understand the range of valid numbers. The number line not only shows the boundary clearly but also the infinite collection of numbers that satisfy the inequality condition. This makes identifying and conceptualizing solutions simpler and more intuitive.