Write a compound inequality that the graph could represent?
$x>-5$ or $x \geq 3$
$x \geq-5$ or $x>3$
$-5 \leq x<3$
$3 \leq x<-5$
Answer (2)
Analyze each compound inequality.
-5"> x > − 5 or x ≥ 3 is equivalent to -5"> x > − 5 .
x ≥ − 5 or 3"> x > 3 is equivalent to x ≥ − 5 .
− 5 ≤ x < 3 means x is between -5 (inclusive) and 3 (exclusive).
3 ≤ x < − 5 is impossible.
Without a graph, we can't definitively choose one inequality, but we have analyzed each option.
Explanation
Analyze the problem We are asked to identify a compound inequality from the given options. Let's analyze each option to determine its meaning.
Analyze option 1
-5"> x > − 5 or x ≥ 3 : This means x is greater than -5, or x is greater than or equal to 3. Since any number greater than or equal to 3 is also greater than -5, this is equivalent to -5"> x > − 5 .
Analyze option 2
x ≥ − 5 or 3"> x > 3 : This means x is greater than or equal to -5, or x is greater than 3. Since any number greater than 3 is also greater than or equal to -5, this is equivalent to x ≥ − 5 .
Analyze option 3
− 5 ≤ x < 3 : This means x is greater than or equal to -5 and less than 3.
Analyze option 4
3 ≤ x < − 5 : This means x is greater than or equal to 3 and less than -5. This is impossible, as no number can satisfy both conditions simultaneously.
Conclusion Without a graph, we can't definitively choose one inequality. However, we can analyze what each inequality represents. Option 1, -5"> x > − 5 , represents all numbers greater than -5. Option 2, x ≥ − 5 , represents all numbers greater than or equal to -5. Option 3, − 5 ≤ x < 3 , represents all numbers between -5 (inclusive) and 3 (exclusive). Option 4 is impossible.
Examples
Understanding compound inequalities is crucial in various real-world scenarios. For instance, consider a thermostat that turns on the heating system when the temperature is below 18 degrees Celsius or turns on the cooling system when the temperature is above 25 degrees Celsius. This can be represented as a compound inequality: T < 18 or 25"> T > 25 , where T is the temperature. Similarly, in quality control, a product might be accepted if its dimensions fall within a certain range, such as 10 ≤ x ≤ 12 cm, where x is the dimension of the product. These examples illustrate how compound inequalities are used to define conditions and ranges in everyday situations.
The suitable compound inequality from the options is − 5 ≤ x < 3 , which represents values between -5 and 3, including -5 but not 3. Options 1 and 2 simplify to broader conditions, while option 4 is impossible. Therefore, option 3 is the only valid representation of a range.
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