Question

exit ticket for ma.7.ar.2.1
exit ticket: solving & graphing inequalities
name: jordyn israel date: 10.23.26

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 interpretations: 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:

<pre_analysis>
{
"quality": "clear",
"question_count": 6,
"skills_matched": [
"step_cot",
"explain_then_answer",
"quick"
],
"needs_drawing": true,
"concepts_used": [
"Solving Inequalities",
"Graphing Inequalities",
"Modeling Inequalities"
],
"new_concepts": [],
"current_concepts": [
"Solving Inequalities",
"Graphing Inequalities",
"Modeling Inequalities"
]
}
</pre_analysis>

<reasoning>

Rotate the image content mentally

The image is upside down. Let's read and solve each question in order from 1 to 6.

Solve Question 1

Solve \(x + 5 < 12\):
\[
x < 12 - 5 \implies x < 7
\]

Solve Question 2

Solve \(-3x \ge 15\). Divide by \(-3\) and flip the inequality sign:
\[
x \le \frac{15}{-3} \implies x \le -5
\]

Solve Question 3

Solve \(2x - 7 > 1\):
\[
2x > 8 \implies x > 4
\]

Solve Question 4

Solve \(\frac{x}{-2} < 4\). Multiply by \(-2\) and flip the inequality sign:
\[
x > -8
\]
The student's claim \(x < -8\) is incorrect because they forgot to flip the inequality sign when multiplying by a negative number.

Solve Question 5

Let \(t\) be the number of tickets. Each ticket costs \(\$8\), and the total spend must be more than \(\$50\):
\[
\text{Inequality: } 8t > 50
\]
\[
\text{Solution: } t > 6.25
\]
Since tickets must be integers, the minimum number of tickets is \(7\).

Solve Question 6

A closed circle on \(-2\) means \(\ge\) or \(\le\). An arrow pointing to the right means greater than or equal to (\(\ge\)).
Therefore, the inequality is \(x \ge -2\).
Looking at the options:

• First option: \(x \ge -2\)
• Second option: \(x \le -2\)
• Third option: \(x > -2\)
• Fourth option: \(x < -2\)

The correct option is \(x \ge -2\).
</reasoning>

<answer>

Question 1

\(x < 7\)

Question 2

\(x \le -5\)

Question 3

\(x > 4\)

Question 4

No, they are incorrect. When multiplying or dividing both sides of an inequality by a negative number, the inequality sign must be flipped. The correct solution is \(x > -8\).

Question 5

• Inequality: \(8t > 50\)
• Solution: \(t > 6.25\) (minimum of \(7\) tickets)

Question 6

\(x \ge -2\)
</answer>

<plot>
{
"elements": [
{
"type": "line",
"params": [[-5, 0], [5, 0]],
"properties": {
"strokeColor": "#583C87",
"strokeWidth": 2
}
},
{
"type": "point",
"params": [[-2, 0]],
"properties": {
"name": "-2",
"size": 4,
"color": "#8C55F2",
"withLabel": true
}
},
{
"type": "bentinequality",
"params": [{ "x": -2, "operator": ">=", "labelText": "x \ge -2" }],
"properties": {
"strokeColor": "#8C55F2",
"strokeWidth": 4
}
}
]
}
</plot>

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

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 6,
"skills_matched": [
"step_cot",
"explain_then_answer",
"quick"
],
"needs_drawing": true,
"concepts_used": [
"Solving Inequalities",
"Graphing Inequalities",
"Modeling Inequalities"
],
"new_concepts": [],
"current_concepts": [
"Solving Inequalities",
"Graphing Inequalities",
"Modeling Inequalities"
]
}
</pre_analysis>

<reasoning>

Rotate the image content mentally

The image is upside down. Let's read and solve each question in order from 1 to 6.

Solve Question 1

Solve \(x + 5 < 12\):
\[
x < 12 - 5 \implies x < 7
\]

Solve Question 2

Solve \(-3x \ge 15\). Divide by \(-3\) and flip the inequality sign:
\[
x \le \frac{15}{-3} \implies x \le -5
\]

Solve Question 3

Solve \(2x - 7 > 1\):
\[
2x > 8 \implies x > 4
\]

Solve Question 4

Solve \(\frac{x}{-2} < 4\). Multiply by \(-2\) and flip the inequality sign:
\[
x > -8
\]
The student's claim \(x < -8\) is incorrect because they forgot to flip the inequality sign when multiplying by a negative number.

Solve Question 5

Let \(t\) be the number of tickets. Each ticket costs \(\$8\), and the total spend must be more than \(\$50\):
\[
\text{Inequality: } 8t > 50
\]
\[
\text{Solution: } t > 6.25
\]
Since tickets must be integers, the minimum number of tickets is \(7\).

Solve Question 6

A closed circle on \(-2\) means \(\ge\) or \(\le\). An arrow pointing to the right means greater than or equal to (\(\ge\)).
Therefore, the inequality is \(x \ge -2\).
Looking at the options:

• First option: \(x \ge -2\)
• Second option: \(x \le -2\)
• Third option: \(x > -2\)
• Fourth option: \(x < -2\)

The correct option is \(x \ge -2\).
</reasoning>

<answer>

Question 1

\(x < 7\)

Question 2

\(x \le -5\)

Question 3

\(x > 4\)

Question 4

No, they are incorrect. When multiplying or dividing both sides of an inequality by a negative number, the inequality sign must be flipped. The correct solution is \(x > -8\).

Question 5

• Inequality: \(8t > 50\)
• Solution: \(t > 6.25\) (minimum of \(7\) tickets)

Question 6

\(x \ge -2\)
</answer>

<plot>
{
"elements": [
{
"type": "line",
"params": [[-5, 0], [5, 0]],
"properties": {
"strokeColor": "#583C87",
"strokeWidth": 2
}
},
{
"type": "point",
"params": [[-2, 0]],
"properties": {
"name": "-2",
"size": 4,
"color": "#8C55F2",
"withLabel": true
}
},
{
"type": "bentinequality",
"params": [{ "x": -2, "operator": ">=", "labelText": "x \ge -2" }],
"properties": {
"strokeColor": "#8C55F2",
"strokeWidth": 4
}
}
]
}
</plot>

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