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Chapter 1: Problem 53

Solve the inequality. Graph the solution. $$ \frac{z}{4} \leq 12 $$

Short Answer

Expert verified
The solution to the inequality is \( z \leq 48 \) and this is represented graphically by highlighting to the left of and including 48 on the number line

Step by step solution

01

Isolate the Variable

In order to isolate the variable \( z \), we need to multiply both sides of the inequality by \( 4 \). This reasoning is based on the principle of the equivalence, which states that if equality is held between two quantities, then the same value can be added, or subtracted, or multiplied, or divided on both sides. By doing so, the inequality remains valid. Thus, we obtain \( z \leq 12 \times 4 \).
02

Perform the Multiplication

The next step is to perform the multiplication on the right side of the inequality: 12 times 4 equals 48. So, the inequality is \( z \leq 48 \).
03

Graphical Representation

For the graphical representation of the solution, imagine or draw a number line. As the answer is \( z \leq 48 \), mark 48 on the number line and color or highlight to the left of it (including 48, as the inequality is less than or equal to 48).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graphical Representation
Graphical representation of inequalities helps us to visualize solutions and understand which numbers satisfy the condition. For the inequality \( z \leq 48 \), you start by sketching a number line.

  • Find the number 48 on the line; this is your critical point.
  • Since the inequality is "less than or equal to," we will use a closed circle on 48. This shows that 48 is included in the solution set.
  • Shade or highlight all numbers to the left of 48, indicating that these values satisfy the inequality.
By creating this visual tool, it becomes clear that any number less than or equal to 48 is a solution to the inequality. This method gives a clear pictorial view of the range of solutions.
Number Line
Using a number line is a basic yet powerful technique in mathematics to represent solutions to inequalities or equations. It serves as a visual tool for understanding which numbers satisfy a given inequality.

For instance, take the inequality \( z \leq 48 \):
  • Draw a horizontal line, marking evenly spaced numbers around the region of interest.
  • Place a circle on the number 48 to emphasize your endpoint. If the inequality were strict (\(<\)), you'd use an open circle to show 48 is not included.
  • Shade the entire line extending from the left and left of 48, beginning from the circle. This shading will encompass all numbers that are possible solutions.
This approach is simple yet effective for students to grasp the notion of which values make an inequality true. It promotes a stronger intuitive understanding.
Multiplication Principle
The multiplication principle is a foundational concept in solving equations and inequalities, stating that you can multiply both sides by the same number without changing the inequality's direction.

In the inequality \( \frac{z}{4} \leq 12 \):
  • We multipled both sides by 4 (the denominator of \( z/4 \)) to isolate \( z \).
  • The multiplication yields \( z \leq 48 \); here, multiplying by a positive number retains the direction of the inequality.
  • If the multiplication involved a negative number, you would have to flip the inequality sign.
This principle helps simplify expressions and solve inequalities efficiently, providing a systematic way to tackle algebraic operations. Always remember, maintaining the balance of the equation is key, and understanding how operations affect inequality is crucial.

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