Question

exit ticket for ma.7.ar.2.1

exit ticket: solving & graphing inequalities
name: ____ date: ____

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 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>

Solve the first inequality

Solving Inequalities
Subtract \(5\) from both sides of the inequality:
\[
x + 5 < 12 \implies x < 7
\]

Solve the second inequality

Solving Inequalities
Divide both sides by \(-3\) and reverse the inequality sign:
\[
-3x \ge 15 \implies x \le -5
\]

Solve the third inequality

Solving Inequalities
Add \(7\) to both sides, then divide by \(2\):
\[
2x - 7 > 1 \implies 2x > 8 \implies x > 4
\]

Analyze the student's claim

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

Model the real-world scenario

Modeling Inequalities
Write the inequality where the total cost of \(t\) tickets at \(\$8\) each is more than \(\$50\):
\[
8t > 50
\]
Solve for \(t\):
\[
t > \frac{50}{8} \implies t > 6.25
\]

Interpret the graph description

Graphing Inequalities
A closed circle on \(-2\) represents \(\ge\) or \(\le\). An arrow pointing to the right represents values greater than or equal to \(-2\):
\[
x \ge -2
\]
This corresponds to option D.
</reasoning>

<answer>

Question 1

\(x < 7\)

Question 2

\(x \le -5\)

Question 3

\(x > 4\)

Question 4

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

Question 5

• Inequality: \(8t > 50\)
• Solution: \(t > 6.25\)

Question 6

<mcq-option>A) \(x < -2\)</mcq-option>
<mcq-option>B) \(x > -2\)</mcq-option>
<mcq-option>C) \(x \le -2\)</mcq-option>
<mcq-correct>D) \(x \ge -2\)</mcq-correct>
</answer>

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

Answer:

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</pre_analysis>

<reasoning>

Solve the first inequality

Solving Inequalities
Subtract \(5\) from both sides of the inequality:
\[
x + 5 < 12 \implies x < 7
\]

Solve the second inequality

Solving Inequalities
Divide both sides by \(-3\) and reverse the inequality sign:
\[
-3x \ge 15 \implies x \le -5
\]

Solve the third inequality

Solving Inequalities
Add \(7\) to both sides, then divide by \(2\):
\[
2x - 7 > 1 \implies 2x > 8 \implies x > 4
\]

Analyze the student's claim

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

Model the real-world scenario

Modeling Inequalities
Write the inequality where the total cost of \(t\) tickets at \(\$8\) each is more than \(\$50\):
\[
8t > 50
\]
Solve for \(t\):
\[
t > \frac{50}{8} \implies t > 6.25
\]

Interpret the graph description

Graphing Inequalities
A closed circle on \(-2\) represents \(\ge\) or \(\le\). An arrow pointing to the right represents values greater than or equal to \(-2\):
\[
x \ge -2
\]
This corresponds to option D.
</reasoning>

<answer>

Question 1

\(x < 7\)

Question 2

\(x \le -5\)

Question 3

\(x > 4\)

Question 4

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

Question 5

• Inequality: \(8t > 50\)
• Solution: \(t > 6.25\)

Question 6

<mcq-option>A) \(x < -2\)</mcq-option>
<mcq-option>B) \(x > -2\)</mcq-option>
<mcq-option>C) \(x \le -2\)</mcq-option>
<mcq-correct>D) \(x \ge -2\)</mcq-correct>
</answer>

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"question_type": "Multi-part",
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