Problem 17 - Algebra 2

Question

Solve each inequality. Then graph the solution set on a number line. \(14>7 y-21\)

Step-by-Step Solution

Answer

The solution is \(y < 5\); on a number line, shade leftwards from an open circle at 5.

Step 1: Add 21 to Both Sides

To eliminate the constant term on the right side of the inequality, we add 21 to both sides: \[14 + 21 > 7y - 21 + 21\]This simplifies to:\[35 > 7y\]

Step 2: Divide Both Sides by 7

To solve for \(y\), we divide each side of the inequality by 7:\[\frac{35}{7} > \frac{7y}{7}\]Simplifying gives:\[5 > y\]This can be rewritten as:\[y < 5\]

Step 3: Graph the Solution on a Number Line

Draw a number line and use an open circle to indicate that 5 is not included in the solution set since the inequality is '<' (strictly less than). Shade the portion of the line to the left of 5 to represent all values that are less than 5.

Key Concepts

Graphing

Graphing is a visual representation of mathematical expressions, in this case, inequalities. It helps you understand the range of possible solutions. When graphing inequalities like the one given, where the inequality is "\(y < 5\)", it is essential to clearly indicate that 5 is not a part of the solution set.
To graph this on a number line, start by drawing a horizontal line. This line represents all possible values of \(y\). Next, locate the number 5 on this line. Since 5 is not included, draw an open circle around it. This open circle is a visual cue that 5 itself is not a solution, keeping the inequality strictly 'less than'.
Now, shade the area to the left of the open circle to demonstrate that all numbers less than 5 satisfy the inequality. Remember, shading is just as important as the open circle. Together, they show which values \(y\) can take in this inequality.

Number Line

A number line is a straight line where numbers are placed at intervals along its length, with each point on the line corresponding to a number.
In solving inequalities, a number line serves as a perfect tool for visualizing the range of solutions. In this context, it helps illustrate not just singular solutions but a spectrum of them along the line.
When the inequality \(y < 5\) is presented on a number line, an open circle at 5 signifies that 5 itself is not included in the solution set. The part of the line to the left of this point is shaded, indicating all potential values of \(y\) that are less than 5.

• The open circle represents the boundary that is not part of the solution.
• Shading to the left shows all the valid solutions for \(y\).

In solving and representing inequalities, the number line is indispensable because it translates the abstract concept into something palpably clear and concise.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions to solve for unknown variables. When solving inequalities, these manipulations must be done with care to maintain the direction of the inequality.
Consider the inequality given \(14 > 7y - 21\). The first step in the solution involves eliminating the constant term by adding 21 to both sides. Doing so preserves the balance of the inequality and simplifies it to \(35 > 7y\).
The next crucial step is to isolate \(y\) by dividing each side of the inequality by 7. This step requires attention, as dividing or multiplying both sides of an inequality by a negative number would reverse the inequality sign, but here we divide by a positive number, so the inequality remains \(5 > y\).

• Add or subtract terms from both sides to balance the equation.
• When multiplying or dividing, consider the sign of the number to maintain the inequality correctly.

This results in \(y < 5\), clearly indicating which values satisfy the original inequality, even before graphing on a number line.