Welcome to Westonci.ca, the place where your questions are answered by a community of knowledgeable contributors. Get immediate answers to your questions from a wide network of experienced professionals on our Q&A platform. Get immediate and reliable solutions to your questions from a community of experienced professionals on our platform.
Sagot :
### Part A: Solve the Inequality
We start with the given inequality:
[tex]\[ \left| \frac{1}{4}x - 2 \right| - 3 \geq 4 \][/tex]
Step 1: Isolate the absolute value expression.
[tex]\[ \left| \frac{1}{4} x - 2 \right| - 3 \geq 4 \][/tex]
Add 3 to both sides:
[tex]\[ \left| \frac{1}{4} x - 2 \right| \geq 7 \][/tex]
Step 2: Translate the absolute value inequality to two separate inequalities.
The absolute value inequality [tex]\(\left| A \right| \geq B\)[/tex] translates to:
[tex]\[ A \geq B \quad \text{or} \quad A \leq -B \][/tex]
In our case, [tex]\(A = \frac{1}{4} x - 2\)[/tex] and [tex]\(B = 7\)[/tex]. Therefore, we write:
[tex]\[ \frac{1}{4} x - 2 \geq 7 \quad \text{or} \quad \frac{1}{4} x - 2 \leq -7 \][/tex]
Step 3: Solve each inequality separately.
First inequality:
[tex]\[ \frac{1}{4} x - 2 \geq 7 \][/tex]
Add 2 to both sides:
[tex]\[ \frac{1}{4} x \geq 9 \][/tex]
Multiply both sides by 4:
[tex]\[ x \geq 36 \][/tex]
Second inequality:
[tex]\[ \frac{1}{4} x - 2 \leq -7 \][/tex]
Add 2 to both sides:
[tex]\[ \frac{1}{4} x \leq -5 \][/tex]
Multiply both sides by 4:
[tex]\[ x \leq -20 \][/tex]
Step 4: Combine the solutions from the two inequalities.
[tex]\[ x \geq 36 \quad \text{or} \quad x \leq -20 \][/tex]
So, the solution to the inequality is:
[tex]\[ x \geq 36 \quad \text{or} \quad x \leq -20 \][/tex]
### Part B: Describe the Graph of the Solution
The graph will represent the intervals where the inequality holds true. We have two intervals on the x-axis:
[tex]\[ (-\infty, -20] \quad \text{and} \quad [36, \infty) \][/tex]
This means that on the x-axis, the solution set includes all values of [tex]\(x\)[/tex] less than or equal to [tex]\(-20\)[/tex], and all values of [tex]\(x\)[/tex] greater than or equal to [tex]\(36\)[/tex].
On the graph, these intervals will be represented by shading:
- The interval [tex]\((- \infty, -20]\)[/tex] will be shaded, including the point [tex]\(-20\)[/tex].
- The interval [tex]\([36, \infty)\)[/tex] will be shaded, including the point [tex]\(36\)[/tex].
Therefore, the graph of the solution will have shaded intervals at [tex]\((- \infty, -20]\)[/tex] and [tex]\([36, \infty)\)[/tex] on the x-axis.
We start with the given inequality:
[tex]\[ \left| \frac{1}{4}x - 2 \right| - 3 \geq 4 \][/tex]
Step 1: Isolate the absolute value expression.
[tex]\[ \left| \frac{1}{4} x - 2 \right| - 3 \geq 4 \][/tex]
Add 3 to both sides:
[tex]\[ \left| \frac{1}{4} x - 2 \right| \geq 7 \][/tex]
Step 2: Translate the absolute value inequality to two separate inequalities.
The absolute value inequality [tex]\(\left| A \right| \geq B\)[/tex] translates to:
[tex]\[ A \geq B \quad \text{or} \quad A \leq -B \][/tex]
In our case, [tex]\(A = \frac{1}{4} x - 2\)[/tex] and [tex]\(B = 7\)[/tex]. Therefore, we write:
[tex]\[ \frac{1}{4} x - 2 \geq 7 \quad \text{or} \quad \frac{1}{4} x - 2 \leq -7 \][/tex]
Step 3: Solve each inequality separately.
First inequality:
[tex]\[ \frac{1}{4} x - 2 \geq 7 \][/tex]
Add 2 to both sides:
[tex]\[ \frac{1}{4} x \geq 9 \][/tex]
Multiply both sides by 4:
[tex]\[ x \geq 36 \][/tex]
Second inequality:
[tex]\[ \frac{1}{4} x - 2 \leq -7 \][/tex]
Add 2 to both sides:
[tex]\[ \frac{1}{4} x \leq -5 \][/tex]
Multiply both sides by 4:
[tex]\[ x \leq -20 \][/tex]
Step 4: Combine the solutions from the two inequalities.
[tex]\[ x \geq 36 \quad \text{or} \quad x \leq -20 \][/tex]
So, the solution to the inequality is:
[tex]\[ x \geq 36 \quad \text{or} \quad x \leq -20 \][/tex]
### Part B: Describe the Graph of the Solution
The graph will represent the intervals where the inequality holds true. We have two intervals on the x-axis:
[tex]\[ (-\infty, -20] \quad \text{and} \quad [36, \infty) \][/tex]
This means that on the x-axis, the solution set includes all values of [tex]\(x\)[/tex] less than or equal to [tex]\(-20\)[/tex], and all values of [tex]\(x\)[/tex] greater than or equal to [tex]\(36\)[/tex].
On the graph, these intervals will be represented by shading:
- The interval [tex]\((- \infty, -20]\)[/tex] will be shaded, including the point [tex]\(-20\)[/tex].
- The interval [tex]\([36, \infty)\)[/tex] will be shaded, including the point [tex]\(36\)[/tex].
Therefore, the graph of the solution will have shaded intervals at [tex]\((- \infty, -20]\)[/tex] and [tex]\([36, \infty)\)[/tex] on the x-axis.
We hope this information was helpful. Feel free to return anytime for more answers to your questions and concerns. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Westonci.ca is here to provide the answers you seek. Return often for more expert solutions.
