Problem 16 - Algebra 2

Question

Solve each inequality. Then graph the solution set on a number line. \(13-4 k \leq 27\)

Step-by-Step Solution

Answer

The solution is \(k \geq -3.5\) and is represented as a solid circle on -3.5 with shading to the right on the number line.

Step 1: Isolate the Variable Term

Subtract 13 from both sides of the inequality to get the variable term by itself. This gives us:\[ -4k \leq 27 - 13 \]Simplifying the right-hand side, we have:\[ -4k \leq 14 \]

Step 2: Solve for the Variable

To isolate \(k\), divide both sides of the inequality by -4. Remember that dividing or multiplying both sides of an inequality by a negative number reverses the inequality sign:\[ k \geq \frac{14}{-4} \]Simplify the fraction:\[ k \geq -3.5 \]

Step 3: Graph the Solution on a Number Line

Mark \(-3.5\) on the number line. Since the inequality is \( k \geq -3.5 \), draw a solid circle at \(-3.5\) to indicate \(-3.5\) is included in the solution set. Then, shade the region to the right of \(-3.5\) as this shows that all numbers greater than or equal to \(-3.5\) satisfy the inequality.

Key Concepts

graphing inequalities on a number line

When graphing inequalities on a number line, it's important to visually represent the range of solutions. For an inequality such as \( k \geq -3.5 \), you first mark \(-3.5\) on the number line. Since \(-3.5\) is included in the solution set, use a solid circle to denote that this point is part of the solution. The inequality symbol \( \geq \) tells us that all numbers greater than or equal to \(-3.5\) are solutions, so you will shade the number line to the right of \(-3.5\).

Here's how to remember:

• If the inequality includes the equal sign (\( \geq \) or \( \leq \)), use a solid circle.
• If it doesn't include the equal sign (\( > \) or \( < \)), use an open circle.
• Shade to the right for greater than, and to the left for less than.

This visual representation makes understanding parts of the number line that satisfy the inequality much easier.

isolating a variable

Isolating the variable involves rearranging the inequality so that the variable is on one side by itself. This step makes it easier to solve for the variable. In our example with the inequality \(13 - 4k \leq 27\), we start by isolating \(-4k\).

The steps include:

• Subtract 13 from both sides: \(-4k \leq 14\)
• Then, to isolate \(k\), you'll need to divide both sides by \(-4\). Remember, dividing or multiplying by a negative number reverses the inequality sign.

So, you get \(k \geq -3.5\).

It's crucial to always consider the operation you apply to each side of an inequality, especially with negative numbers. Isolating the variable neatly sets the stage for finding the solution set.

properties of inequalities

Understanding the properties of inequalities helps in successfully solving them. Just like equations, inequalities come with a set of rules:

• **Addition or Subtraction**: You can add or subtract the same number from both sides of an inequality without changing the direction of the inequality.
• **Multiplication or Division by a Positive Number**: Doing this keeps the inequality sign the same.
• **Multiplying or Dividing by a Negative Number**: This is the critical difference from equations. Multiplying or dividing both sides by a negative number reverses the inequality sign (\(<\) becomes \(>\) and vice versa).

These properties derive from logical reasoning about order and balance, and taking note of them is key when working through inequality problems.

inequality solution sets

The solution set of an inequality includes all the values that make the inequality true. For the inequality \(k \geq -3.5\), the solution set is all numbers greater than or equal to \(-3.5\). This collection of solutions can often be infinite.

When you express the solution set:

• In number line notation, mark the critical point with a circle (solid or open depending on the sign).
• In interval notation, this solution set is represented as \([-3.5, \infty)\).

Being able to identify and express solution sets comprehensively helps in communicating the range of possible solutions. Visualization through graphs and precise notation contributes to a clearer understanding of what the inequality represents.