Problem 8 - Prealgebra

Question

Solve each inequality and check your solution. Then graph the solution on a number line. $$-2(k+1) \geq 16$$

Step-by-Step Solution

Answer

The solution is \(k \leq -9\).

Step 1: Distribute the Coefficient

First, distribute the -2 to both terms inside the parenthesis. This means you multiply -2 by both \(k\) and 1. So we have:\[-2 \cdot (k+1) = -2k - 2\] This simplifies our inequality to:\[-2k - 2 \geq 16\]

Step 2: Isolate the Variable Term

Next, add 2 to both sides of the inequality to move the constant term:\[-2k - 2 + 2 \geq 16 + 2\]This simplifies to:\[-2k \geq 18\]

Step 3: Solve for the Variable

To solve for \(k\), divide both sides by -2, remembering to flip the inequality sign:\[\frac{-2k}{-2} \leq \frac{18}{-2}\]Resulting in:\[k \leq -9\]

Step 4: Check the Solution

To check, choose a value less than or equal to -9 and substitute back into the original inequality. Use \(k = -10\) as an example:\[-2(-10 + 1) = -2(-9) = 18\]Since \(18 \geq 16\), the solution \(k \leq -9\) is confirmed.

Step 5: Graph the Solution

Draw a number line, mark the point -9, and shade all values to the left, including -9, to represent \(k \leq -9\). Use a closed circle on -9 to indicate that it is part of the solution.

Key Concepts

Distributive Property

In mathematics, the distributive property is a fundamental concept that allows you to simplify expressions by distributing a factor across terms inside parentheses. In our exercise, the expression \(-2(k+1)\) requires using the distributive property.

• Multiply \(-2\) by each term inside the parentheses: \(k\) and \(1\).
• This gives us: \(-2 \cdot k = -2k\) and \(-2 \cdot 1 = -2\).
• Combine these results to rewrite the expression as \(-2k - 2\).

This application simplifies the inequality, making it easier to solve. Using the distributive property effectively allows you to manipulate and simplify equations before proceeding with further steps.

Number Line

A number line is a visual representation of numbers laid out on a straight line. For inequalities, it helps illustrate the range of possible solutions.

• After solving the inequality \(k \leq -9\), the number line is used to graph the solution.
• Mark the point \(-9\) on the line with a closed circle, since \(-9\) is included in the solution set.
• Shade the entire region to the left of \(-9\), indicating all values less than or equal to \(-9\).

The number line is beneficial, as it provides a clear, visual understanding of the solution set for the inequality, making it easier to conceptualize the range of valid solutions.

Checking Solutions

Checking solutions is a crucial step when solving inequalities to ensure correctness. It involves substituting a possible solution back into the original inequality to verify its validity.

• For \(k \leq -9\), choose a test value, such as \(k = -10\).
• Substitute \(-10\) into the original inequality: \(-2(-10 + 1)\).
• Simplify to find \(18\), which satisfies the condition \(18 \geq 16\).

This confirms that the assessed solution is correct. This step guards against mistakes and ensures confidence in your work.

Solving Inequalities

Solving inequalities involves finding the set of values that satisfy a given inequality. It is similar to solving equations, but requires special attention to how the inequality behaves, especially when multiplying or dividing by negative numbers.

• Firstly, use methods such as the distributive property to simplify the inequality.
• Isolate the variable by moving all terms involving the variable to one side and constants to the other.
• In this example, adding 2 to both sides gives \(-2k \geq 18\). Divide by \(-2\), and remember to flip the inequality sign, resulting in \(k \leq -9\).

Carefully solving inequalities requires a good understanding of how operations affect the inequality, especially when dealing with negative numbers. It is essential to always verify your solution.