Question

exit ticket for ma.7.ar.2.1
exit ticket: solving & graphing inequalities
name: jordyn manuel date: 1.23.26

1. one-step basics: solve the inequality and graph the solution on a number line:

\\x + 5 < 12\\

1. the \flip\ rule: solve the inequality below. hint: watch your sign!

\\-3x \ge 15\\

1. two-step challenge: solve the following inequality:

\\2x - 7 > 1\\

1. multiplication & division: a student is solving \\(\frac{x}{-2} < 4\\). they claim the answer is \\(x < -8\\). are they correct? explain your reasoning.
2. real-world modeling: to qualify for a discount, a group must spend more than \\(\\$50\\). if each ticket costs \\(\\$8\\), write and solve an inequality to find the minimum number of tickets (t) the group must buy.

• inequality:
• solution: \\(t >\\)

1. graph interpretation: which inequality is represented by a number line with a closed circle on \\(-2\\) and an arrow pointing to the right?

• a) \\(x < -2\\)
• b) \\(x > -2\\)
• c) \\(x \le -2\\)
• d) \\(x \ge -2\\)

Explanation:

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<reasoning>

Rotate the image content mentally

The image is upside down. Let's read the questions by rotating them 180 degrees:

Title: Exit Ticket for MA.7.AR.2.1
Exit Ticket: Solving & Graphing Inequalities

1. One-Step Basics: Solve the inequality and graph the solution on a number line:

\(x + 5 < 12\)

1. The "Flip" Rule: Solve the inequality below. Hint: Watch your sign!

\(-3x \ge 15\)

1. Two-Step Challenge: Solve the following inequality:

\(2x - 7 > 1\)

1. Multiplication & Division: A student is solving \(\frac{x}{-2} < 4\). They claim the answer is \(x < -8\). Are they correct? Explain your reasoning.

1. Real-World Modeling: To qualify for a discount, a group must spend more than $50. If each ticket costs $8, write and solve an inequality to find the minimum number of tickets (\(t\)) the group must buy.

• Inequality: _________
• Solution: \(t >\) _________

1. Graph Interpretations: Which inequality is represented by a number line with a closed circle on \(-2\) and an arrow pointing to the right?

• \(x < -2\)
• \(x > -2\)
• \(x \le -2\)
• \(x \ge -2\)

Solve Question 1

\[

$$\begin{aligned} x + 5 &< 12 \\ x &< 7 \end{aligned}$$

\]

Solve Question 2

\[

$$\begin{aligned} -3x &\ge 15 \\ x &\le -5 \end{aligned}$$

\]

Solve Question 3

\[

$$\begin{aligned} 2x - 7 &> 1 \\ 2x &> 8 \\ x &> 4 \end{aligned}$$

\]

Solve Question 4

No. When multiplying or dividing both sides of an inequality by a negative number, the inequality sign must be flipped.
\[

$$\begin{aligned} \frac{x}{-2} &< 4 \\ x &> -8 \end{aligned}$$

\]

Solve Question 5

\[

$$\begin{aligned} \text{Inequality: } &8t > 50 \\ \text{Solution: } &t > 6.25 \quad (\text{or } t \ge 7 \text{ tickets}) \end{aligned}$$

\]

Solve Question 6

A closed circle on \(-2\) means \(\ge\) or \(\le\). An arrow pointing to the right means greater than or equal to (\(\ge\)).
\[
x \ge -2
\]
</reasoning>

<answer>

No.	Problem / Prompt	Answer
2	Solve \(-3x \ge 15\)	\(x \le -5\)
3	Solve \(2x - 7 > 1\)	\(x > 4\)
4	Is the student's claim \(x < -8\) correct for \(\frac{x}{-2} < 4\)?	No, the inequality sign must flip when multiplying by a negative number; the correct solution is \(x > -8\).
5	Write and solve the inequality for spending more than $50 with $8 tickets	Inequality: \(8t > 50\)<br>Solution: \(t > 6.25\) (minimum of 7 tickets)
6	Which inequality has a closed circle on \(-2\) pointing right?	\(x \ge -2\)

</answer>

<post_analysis>
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"subject": "Mathematics",
"question_type": "Multi-part",
"knowledge_point": [
"Mathematics",
"Algebra",
"Linear Inequalities"
]
}
</post_analysis>

Answer:

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</pre_analysis>

<reasoning>

Rotate the image content mentally

The image is upside down. Let's read the questions by rotating them 180 degrees:

Title: Exit Ticket for MA.7.AR.2.1
Exit Ticket: Solving & Graphing Inequalities

1. One-Step Basics: Solve the inequality and graph the solution on a number line:

\(x + 5 < 12\)

1. The "Flip" Rule: Solve the inequality below. Hint: Watch your sign!

\(-3x \ge 15\)

1. Two-Step Challenge: Solve the following inequality:

\(2x - 7 > 1\)

1. Multiplication & Division: A student is solving \(\frac{x}{-2} < 4\). They claim the answer is \(x < -8\). Are they correct? Explain your reasoning.

1. Real-World Modeling: To qualify for a discount, a group must spend more than $50. If each ticket costs $8, write and solve an inequality to find the minimum number of tickets (\(t\)) the group must buy.

• Inequality: _________
• Solution: \(t >\) _________

1. Graph Interpretations: Which inequality is represented by a number line with a closed circle on \(-2\) and an arrow pointing to the right?

• \(x < -2\)
• \(x > -2\)
• \(x \le -2\)
• \(x \ge -2\)

Solve Question 1

\[

$$\begin{aligned} x + 5 &< 12 \\ x &< 7 \end{aligned}$$

\]

Solve Question 2

\[

$$\begin{aligned} -3x &\ge 15 \\ x &\le -5 \end{aligned}$$

\]

Solve Question 3

\[

$$\begin{aligned} 2x - 7 &> 1 \\ 2x &> 8 \\ x &> 4 \end{aligned}$$

\]

Solve Question 4

No. When multiplying or dividing both sides of an inequality by a negative number, the inequality sign must be flipped.
\[

$$\begin{aligned} \frac{x}{-2} &< 4 \\ x &> -8 \end{aligned}$$

\]

Solve Question 5

\[

$$\begin{aligned} \text{Inequality: } &8t > 50 \\ \text{Solution: } &t > 6.25 \quad (\text{or } t \ge 7 \text{ tickets}) \end{aligned}$$

\]

Solve Question 6

A closed circle on \(-2\) means \(\ge\) or \(\le\). An arrow pointing to the right means greater than or equal to (\(\ge\)).
\[
x \ge -2
\]
</reasoning>

<answer>

No.	Problem / Prompt	Answer
2	Solve \(-3x \ge 15\)	\(x \le -5\)
3	Solve \(2x - 7 > 1\)	\(x > 4\)
4	Is the student's claim \(x < -8\) correct for \(\frac{x}{-2} < 4\)?	No, the inequality sign must flip when multiplying by a negative number; the correct solution is \(x > -8\).
5	Write and solve the inequality for spending more than $50 with $8 tickets	Inequality: \(8t > 50\)<br>Solution: \(t > 6.25\) (minimum of 7 tickets)
6	Which inequality has a closed circle on \(-2\) pointing right?	\(x \ge -2\)

</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multi-part",
"knowledge_point": [
"Mathematics",
"Algebra",
"Linear Inequalities"
]
}
</post_analysis>