1. graph the solutions to $x 2$ on the number line. 2. graph the solutions to $p 3$ has the same solutions as $3 x$. yuna says that $x 3$ has the same solutions as $3

Question

1. graph the solutions to $x > 2$ on the number line. 2. graph the solutions to $p < 7$ on the number line. 3. what are three solutions to the inequality in problem 2? 4. how many solutions to the inequality in problem 2 exist? explain. 5. write two inequalities that each represent the solutions graphed on the number line. use $x$ as the variable. 6. adesh says that $x > 3$ has the same solutions as $3 > x$. yuna says that $x > 3$ has the same solutions as $3 < x$. which of the following statements is true? a. only adesh is correct. b. only yuna is correct. c. both adesh and yuna are correct. d. neither adesh nor yuna is correct.

Accepted Answer

### 1. Graph the solutions to $ x > 2 $ on the number line. # Explanation: ## Step1: Identify the critical point The inequality is $ x > 2 $, so the critical point is $ 2 $. Since the inequality is strict ($ > $, not $ \geq $), we use an open circle at $ 2 $. ## Step2: Determine the direction of the line For $ x > 2 $, the solutions are all numbers greater than $ 2 $, so we draw an arrow to the right of the open circle at $ 2 $ on the number line. # Answer: - Draw an open circle at $ 2 $ on the number line. - Draw an arrow starting from the open circle and pointing to the right (towards numbers greater than $ 2 $, like $ 3, 4, 5, \dots $). ### 2. Graph the solutions to $ p < 7 $ on the number line. # Explanation: ## Step1: Identify the critical point The inequality is $ p < 7 $, so the critical point is $ 7 $. Since the inequality is strict ($ < $, not $ \leq $), we use an open circle at $ 7 $. ## Step2: Determine the direction of the line For $ p < 7 $, the solutions are all numbers less than $ 7 $, so we draw an arrow to the left of the open circle at $ 7 $ on the number line. # Answer: - Draw an open circle at $ 7 $ on the number line. - Draw an arrow starting from the open circle and pointing to the left (towards numbers less than $ 7 $, like $ 6, 5, 4, \dots $). ### 3. What are three solutions to the inequality in problem 2? # Explanation: ## Step1: Recall the inequality The inequality from problem 2 is $ p < 7 $. We need to find three numbers that are less than $ 7 $. ## Step2: Choose three numbers Any numbers less than $ 7 $ are valid. For example, $ 6 $, $ 5 $, and $ 0 $ (or $ 3 $, $ -2 $, $ 6.9 $, etc.). # Answer: Three solutions to $ p < 7 $ are $ 6 $, $ 5 $, and $ 0 $ (answers may vary; any numbers less than $ 7 $ are correct). ### 4. How many solutions to the inequality in problem 2 exist? Explain. # Explanation: ## Step1: Recall the inequality The inequality is $ p < 7 $. ## Step2: Analyze the set of solutions The solutions to $ p < 7 $ include all real numbers (integers, fractions, decimals) that are less than $ 7 $. For example, $ 6, 5, 0, -1, 6.9, \frac{1}{2}, -100, \dots $. There are infinitely many real numbers less than $ 7 $ because between any two distinct numbers less than $ 7 $, there are more numbers (e.g., between $ 6 $ and $ 7 $, there are $ 6.1, 6.2, 6.11, 6.12, \dots $, and this pattern continues infinitely). # Answer: There are infinitely many solutions to $ p < 7 $. Explanation: The inequality $ p < 7 $ includes all real numbers (integers, fractions, decimals) that are less than $ 7 $. Since there are infinitely many real numbers less than $ 7 $ (e.g., all integers like $ 6, 5, 4, \dots $, all fractions like $ \frac{1}{2}, \frac{3}{4}, \dots $, and all decimals like $ 6.1, 6.2, 6.11, \dots $ between $ 6 $ and $ 7 $, and negative numbers), the number of solutions is infinite. ### 5. Write two inequalities that each represent the solutions graphed on the number line. Use $ x $ as the variable. # Explanation: ## Step1: Analyze the number line graph The number line has a closed (filled) or open circle? Wait, looking at the graph: the number line has an open circle at $ -1 $ and the line is shaded to the left (towards numbers less than $ -1 $). Wait, the graph shows: from $ -10 $ to $ -1 $, shaded, with an open circle at $ -1 $. So the solutions are all numbers less than $ -1 $. ## Step2: Write two inequalities One inequality is $ x < -1 $. Another equivalent inequality can be written by reversing the inequality and flipping the sign, but since it's a strict inequality (open circle), we can also write $ x \leq -2 $ (but wait, no—wait, the shaded region is all numbers less than $ -1 $. Wait, actually, the graph is: open circle at $ -1 $, shaded to the left (so $ x < -1 $). Another way: we can write $ x + 1 < 0 $ (since $ x < -1 $ is equivalent to $ x + 1 < 0 $). Or $ 2x < -2 $ (dividing both sides by $ 2 $ gives $ x < -1 $). Wait, let's check: The number line has an open circle at $ -1 $ and is shaded to the left, so the solution is $ x < -1 $. To write another inequality, we can manipulate $ x < -1 $. For example, add $ 2 $ to both sides: $ x + 2 < -1 + 2 $ → $ x + 2 < 1 $. Or multiply both sides by $ 3 $ (positive, so inequality sign remains): $ 3x < -3 $. So two inequalities: $ x < -1 $ and $ 3x < -3 $ (or $ x + 2 < 1 $, etc.). # Answer: Two inequalities representing the solutions (all numbers less than $ -1 $) are $ \boldsymbol{x < -1} $ and $ \boldsymbol{3x < -3} $ (answers may vary; other valid inequalities include $ x + 1 < 0 $, $ 2x < -2 $, etc.). ### 6. Adesh says that $ x > 3 $ has the same solutions as $ 3 > x $. Yuna says that $ x > 3 $ has the same solutions as $ 3 < x $. Which of the following statements is true? # Explanation: ## Step1: Analyze Adesh’s statement Adesh says $ x > 3 $ (solutions: $ x $ is greater than $ 3 $) is the same as $ 3 > x $ (solutions: $ x $ is less than $ 3 $). These are not the same because $ x > 3 $ includes $ 4, 5, 6, \dots $, while $ 3 > x $ includes $ 2, 1, 0, \dots $. So Adesh is incorrect. ## Step2: Analyze Yuna’s statement Yuna says $ x > 3 $ is the same as $ 3 < x $. By the definition of inequalities, $ x > 3 $ and $ 3 < x $ are equivalent (they both mean $ x $ is greater than $ 3 $; the inequality $ a > b $ is the same as $ b < a $). So Yuna is correct. ## Step3: Determine the correct option Since only Yuna is correct, the answer is B. # Answer: B. Only Yuna is correct.

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QUESTION IMAGE

Question

Explanation:

1. Graph the solutions to \( x > 2 \) on the number line.

Step1: Identify the critical point

Step2: Determine the direction of the line

Step1: Identify the critical point

Step2: Determine the direction of the line

Step1: Recall the inequality

Step2: Choose three numbers

Answer:

2. Graph the solutions to \( p < 7 \) on the number line.