Problem 64 - Elementary Algebra Within Reach

Question

Solve and graph the inequality. $$-7(z+4)>14$$

Step-by-Step Solution

Answer

The solution to the inequality is \( z < -6 \). The graph is a number line with an open circle over -6 and an arrow pointing leftwards from the open circle.

Step 1: Distribute multiplication

Distribute the multiplication across the parenthesis. This gives \(-7z - 28 > 14\).

Step 2: Isolate the variable z

To isolate z on one side of the inequality, add 28 to both sides of the equation to move -28 from the left side. This results in \(-7z > 42\).

Step 3: Final step in solving the inequality

To find the value of z, divide both sides of the inequality by -7. Remember, when dividing by a negative number, the sign of the inequality flips. Therefore, the final result of the inequality is \(z < -6\).

Step 4: Graphing the inequality

On a number line, a open circle is placed over -6 and an arrow is drawn pointing to the left from the open circle indicating all numbers less than -6.

Key Concepts

Distributive Property

The distributive property is a fundamental algebraic principle that allows you to multiply a number by a sum or difference inside parentheses. In essence, it helps to "distribute" the multiplication across the terms inside the parentheses. Let's consider the inequality

• -7(z + 4) > 14

Using the distributive property, multiply -7 with each term inside the parentheses. Therefore:

• -7(z + 4) = -7z - 28

The result is

• -7z - 28 > 14

This step simplifies the problem and allows us to remove the parentheses, making the inequality easier to work with. Remember, the distributive property is helpful whenever you encounter brackets or parentheses in algebra, and applying it correctly is crucial for solving equations and inequalities.

Isolating Variables

Isolating variables is a critical step when solving inequalities or equations. The goal is to get the variable by itself on one side of the inequality or equation. This involves performing operations that "undo" what is being done to the variable. Let’s use the result from the distributive property:

• -7z - 28 > 14

To isolate the variable z, follow these steps:

• Add 28 to both sides of the inequality to cancel out -28 on the left side, resulting in:
• -7z > 42

Now, divide both sides by -7 to solve for z. It's important to remember that when dividing or multiplying both sides of an inequality by a negative number, you must reverse the inequality sign. In this case, since you divide by -7:

• z < -6

This operation allows us to conclude what range of values z can take to satisfy the original inequality.

Graphing Inequalities

Graphing inequalities on a number line is a visual way to represent the solution set of an inequality. It helps in understanding which values make the inequality true. For our inequality,

• z < -6

Start by drawing a number line and find -6 on it. At -6, place an open circle to indicate that -6 is not included in the solution set. Remember, an open circle is used for inequalities that are strict, such as < or >, where the boundary number itself is not a solution.

• Example: z < -6 indicates all numbers less than -6 are solutions, but not -6 itself.

Next, draw an arrow from the open circle pointing to the left, which shows that the values of z continue indefinitely in that direction. This graphically represents that any number less than -6 satisfies the inequality.