9th Grade
How Do You Create and Solve Inequalities in One Variable Using Multiplication and Division?
{
"voice_prompt": "",
"manuscript": {
"title": {
"text": "How Do You Create and Solve Inequalities in One Variable Using Multiplication and Division?",
"audio": "How do you create and solve inequalities in one variable using multiplication and division?"
},
"description": {
"text": "You can create and solve inequalities in one variable by following these steps: 1. Understand the situation. 2. Write the inequality using the correct comparison symbol. 3. Solve the inequality by isolating the variable. 4. Check your answer.",
"audio": "You can create and solve inequalities in one variable by following these steps: 1. Understand the situation. 2. Write the inequality using the correct comparison symbol. 3. Solve the inequality by isolating the variable. 4. Check your answer."
},
"scenes": [
{
"text": "Inequalities help you describe situations where more than one value can work. You use the symbols less than, greater than, less than or equal to, and greater than or equal to to compare quantities and show a whole range of solutions.",
"latex": "<, >, \\leq, \\geq",
"pop_animation_prompt": "Create pop animations for the following math elements as they are referenced: First, pop the \"<\" symbol when \"less than\" is mentioned. Next, pop the \">\" symbol with \"greater than.\" Then, pop the \"\\leq\" symbol with \"less than or equal to.\" Finally, pop the \"\\geq\" symbol with \"greater than or equal to.\" Ensure each symbol pops at the exact moment it is referenced in the transcript for clarity and emphasis."
},
{
"text": "Let\u2019s start with a real-world example. You\u2019re paid fifteen dollars per hour. You want to earn at least one hundred twenty dollars. How many hours do you need to work?",
"latex": "15 + 120",
"pop_animation_prompt": "Create pop animations for the mathematical expression \"15 + 120\". First, pop the number \"15\" when it is mentioned as \"fifteen\" in the transcript. Next, pop the number \"120\" when it is referenced as \"one hundred twenty\". Ensure each number pops at the exact moment it is mentioned to align with the voiceover, enhancing the viewer's understanding of the example."
},
{
"text": "Let x represent the number of hours you work. Since you want to earn at least one hundred twenty dollars, you write the inequality: fifteen times x is greater than or equal to one hundred twenty.",
"latex": "15x \\geq 120",
"pop_animation_prompt": "Create pop animations for the following math elements as they are referenced in the transcript: First, pop the symbol \"x\" when mentioned as the number of hours worked. Next, pop the number \"120\" when the target earnings are discussed. Then, pop the number \"15\" and the symbol \"x\" together when the inequality is introduced. Finally, pop the operator \"\\(\\geq\\)\" and the number \"120\" together when explaining the inequality's meaning."
},
{
"text": "Now solve the inequality by dividing both sides by fifteen to isolate x. This gives you x is greater than or equal to eight. You need to work at least eight hours.",
"latex": "\\dfrac{15x}{15} \\geq \\dfrac{120}{15} \\quad x \\geq 8",
"pop_animation_prompt": "Create pop animations for the following elements as they are referenced: First, pop the number \"15\" (index 8) when mentioned as \"fifteen\". Next, pop the symbol \"x\" (index 21) when mentioned as \"x\". Finally, pop the number \"8\" (index 23) when mentioned as \"eight\". Ensure each element pops at the exact moment it is referenced in the transcript for clarity and emphasis."
},
{
"text": "Check your answer. Try x equals eight. Fifteen times eight is one hundred twenty \u2014 and that meets the goal. So the solution is correct.",
"latex": "15 \\times 8 = 120",
"pop_animation_prompt": "Create pop animations for the mathematical expression \"15 \\times 8 = 120\" as follows: First, pop the number \"8\" when referenced. Next, pop \"15\", \"\\times\", and \"8\" simultaneously. Finally, pop \"120\" when mentioned. Ensure each element pops at the exact moment it is referenced in the transcript to enhance understanding."
},
{
"text": "Now here\u2019s an example with a negative number. You're descending in an elevator that starts at ground level. Each floor you go down takes you 5 meters lower. You want to go no lower than 25 meters below ground. What's the maximum number of floors you can descend?",
"latex": "-5 - 25",
"pop_animation_prompt": "Create pop animations for the mathematical expression \"-5 - 25\" as follows: First, pop the operator \"-\" and the number \"5\" together when referenced. Next, pop the operator \"-\" and the number \"25\" together when referenced. Ensure the animations highlight the descent in meters as described in the transcript."
},
{
"text": "Let x be the number of floors you descend. Since each floor lowers you by 5 meters, your position is given by negative 5 times x, or \u22125x.",
"latex": "-5x + 1",
"pop_animation_prompt": "Create pop animations for the mathematical expression as follows: First, pop the symbol \"x\" when it is mentioned. Next, pop the number \"5\" when referenced. Then, pop the operator \"-\" when \"negative\" is mentioned. Finally, pop the entire expression \"-5x + 1\" when it is referenced as \"5 times x\" and again as \"\u22125 x\". Ensure each pop aligns with its respective mention in the transcript."
},
{
"text": "You want to stay at or above negative 25 meters, so you write the inequality minus 5 is greater than or equal to minus 25.",
"latex": "-5x \\geq -25",
"pop_animation_prompt": "Create pop animations for the following elements in sequence: First, pop the negative sign and number 25 together when \"negative 25 meters\" is mentioned. Next, pop the negative sign and number 5 together when \"minus 5\" is referenced. Then, pop the greater than or equal to symbol when \"greater than or equal to\" is mentioned. Finally, pop the negative sign and number 25 together again when \"minus 25\" is referenced. Ensure each pop aligns with the corresponding verbal cue in the transcript."
},
{
"text": "Now divide both sides by negative five. Here\u2019s the key rule: whenever you divide or multiply both sides of an inequality by a negative number, you must flip the inequality sign. The inequality becomes x is less than or equal to five.",
"latex": "\\dfrac{-5x}{-5} \\leq \\dfrac{-25}{-5} \\quad x \\leq 5",
"pop_animation_prompt": "Create pop animations for the following elements in sequence: First, pop the denominators \"-5\" at indices 6 and 17 when mentioned. Next, pop the inequality operators \"\\leq\" at indices 9 and 21 when the rule about flipping the inequality is discussed. Finally, pop the symbol \"x\" at index 20, the operator \"\\leq\" at index 21, and the number \"5\" at index 22 when the new inequality is stated. Ensure each pop aligns with its corresponding reference in the transcript for clarity."
},
{
"text": "So the maximum number of floors you can descend without going lower than 25 meters underground is 5. Check your answer. Try x equals five. Negative five times five equals negative twenty-five. That\u2019s still within the limit, so the solution works.",
"latex": "-5 \\times 5 = -25",
"pop_animation_prompt": "Create pop animations for the mathematical expression \"-5 \\times 5 = -25\" as follows: First, pop the number \"25\" when mentioned. Then, simultaneously pop the numbers \"5\" and \"5\" when referenced together. Next, pop the operator \"-\" followed by the number \"5\", then the operator \"\\times\", and the number \"5\" again. Continue with the operator \"=\", followed by the operator \"-\" and the number \"25\". Ensure each element pops at the exact moment it is referenced in the transcript."
},
{
"text": "Flipping the sign makes sure your solution is still correct. If you forget to flip the inequality, your answer might include values that don\u2019t actually make the original inequality true \u2014 and that can lead to incorrect conclusions in real-life situations.",
"latex": "x \\geq 5",
"pop_animation_prompt": "Create pop animations for the mathematical expression. First, pop the operator \"\\(\\geq\\)\" (index 2) when mentioned in the transcript. Simultaneously, pop the operator \"\\(\\leq\\)\" (index 7) as they are referenced together. Ensure both operators pop at the same time to emphasize the concept of flipping inequalities."
}
],
"outro": {
"text": "To solve inequalities using multiplication or division: 1. Understand the problem. 2. Write the inequality. 3. Divide or multiply to isolate the variable. 4. Check that your answer makes the inequality true.",
"audio": "To solve inequalities using multiplication or division: 1. Understand the problem. 2. Write the inequality. 3. Divide or multiply to isolate the variable. 4. Check that your answer makes the inequality true."
}
}
}
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