Problem 23 - Algebra 2

Question

Solve each inequality. Then graph the solution set on a number line. \(\frac{k+7}{3}-1<0\)

Step-by-Step Solution

Answer

The solution is \(k < -4\). Graph: open circle on -4, shade left.

Step 1: Clear the Fraction

To eliminate the fraction, multiply each term by 3, the denominator, to simplify the inequality. This involves multiplying both sides of the inequality by 3:\[3 \left( \frac{k+7}{3} - 1 \right) < 3 \times 0\]Simplifying, we have:\[k + 7 - 3 < 0\]Now, combine like terms:\[k + 4 < 0\]

Step 2: Isolate the Variable

To isolate \(k\), subtract 4 from both sides of the inequality:\[k + 4 - 4 < 0 - 4\]This simplifies to:\[k < -4\]

Step 3: Graph the Solution

To graph the inequality \(k < -4\), draw a number line. Mark a point at \(-4\), and use an open circle at this point to show that \(-4\) is not included in the solution set. Shade the region to the left of \(-4\) to indicate all values less than \(-4\).---Number line representation---•=================<--------------------------------7 -6 -5 -4 -3 -2 -1 0 1 2 3

Key Concepts

Solving Inequalities

Inequalities are similar to equations, but instead of an equal sign, they use symbols like ">", "<", "⩽", and "⩾" to compare two expressions. Solving inequalities follows similar steps as solving equations; however, it requires careful attention to the inequality sign. One of the most important rules when working with inequalities is that if you multiply or divide both sides by a negative number, you need to reverse the inequality sign. Let's delve into the process of solving the given inequality step-by-step.

To solve the inequality \( \frac{k+7}{3}-1<0 \), we begin by eliminating the fraction. Multiply every term by the denominator, which is 3, to simplify the expression. After multiplying, we obtain \( k + 7 - 3 < 0 \). It's important to perform this step carefully to maintain the balance of the inequality.

Next, simplify the expression by combining like terms, which leads us to \( k + 4 < 0 \). The goal is to isolate the variable \( k \), and to achieve this, subtract 4 from both sides. This results in \( k < -4 \).

Through these steps, we have determined that the solution set for the inequality is all real numbers less than -4.

Graphing Inequalities

Graphing inequalities offers a visual representation of the solution set, making it easier to understand which values satisfy the inequality. When graphing the inequality \( k < -4 \), we use a number line to demonstrate the set of possible solutions.

Start by drawing a horizontal line, which is your number line, and clearly label the numbers around the solution. In this case, place a point at -4 for reference. Since the inequality sign is "<" and not "⩽", it tells us that -4 is not part of the solution. Therefore, we use an open circle at -4 to indicate this.

After marking -4, shade the area to the left of -4. This shading implies that all numbers less than -4 satisfy the inequality. By graphing this way, anyone observing the number line can instantly identify that any number to the left of -4 is a viable solution.

Number Lines

Understanding number lines is crucial for visualizing and solving inequalities. A number line is a straight line where every point corresponds to a real number. It provides a graphical way to represent numbers and their relationships.

Number lines are helpful in many ways when working with inequalities. They allow us to:

• Quickly identify which numbers satisfy an inequality.
• Easily compare numbers and determine order and spacing.
• Visually communicate solutions in a clear and straightforward manner.

When graphing the solution \( k < -4 \), we use an open circle because -4 is not included in the solution. The shaded area extending left from -4 signifies that all numbers in this region are solutions. Number lines help students see not just the boundary of the solution but also all numbers encompassed within it, reinforcing a deeper understanding of inequalities.