When ordering integers from least to greatest, you need to understand how negative numbers work on the number line. Negative numbers get smaller as their absolute value increases, while positive numbers get larger as their absolute value increases.

To arrange these integers correctly, visualize them on a number line. Starting from the left (smallest): $$-12$$ is the smallest because it's the farthest from zero in the negative direction. Next comes $$-8$$, then $$-3$$ (closer to zero), then $$0$$, and finally $$5$$ as the largest positive number. This gives us: $$-12, -8, -3, 0, 5$$, which matches choice B.

Choice A places $$-12$$ at the end, treating it as if it were the largest number. This is a common error where students focus on the digit 12 and forget that the negative sign makes it smaller than all other numbers in the list.

Choice C starts with $$0$$ and then lists negative numbers in decreasing order (getting more negative), which is backwards. This shows a misunderstanding of how negative numbers relate to zero and each other.

Choice D arranges the numbers from greatest to least instead of least to greatest, giving the completely opposite order from what's requested.

Remember this key principle: when comparing negative integers, the one with the larger absolute value is actually the smaller number. So $$-12 < -8 < -3 < 0 < 5$$. Always place negative numbers to the left of zero and positive numbers to the right when visualizing the number line.

When you encounter temperature change problems, you need to track each change step by step, being careful with positive and negative values.

Start at 6 AM with $$-8°F$$. By noon, the temperature rose $$15°F$$, so you add: $$-8 + 15 = 7°F$$. From noon to 6 PM, it dropped $$9°F$$, so you subtract: $$7 - 9 = -2°F$$. Finally, from 6 PM to midnight, it warmed $$3°F$$: $$-2 + 3 = 1°F$$.

Looking at the wrong answers: Choice A ($$-5°F$$) likely comes from incorrectly calculating the 6 PM temperature as $$-8°F$$ instead of $$-2°F$$, then adding $$3°F$$ to get $$-5°F$$. This error occurs when students subtract the noon rise from the starting temperature instead of adding it. Choice B ($$-2°F$$) is the temperature at 6 PM, not midnight - this happens when you forget the final step of adding the $$3°F$$ warming from 6 PM to midnight. Choice D ($$4°F$$) results from miscalculating the 6 PM temperature as $$1°F$$ instead of $$-2°F$$, then adding $$3°F$$.

The correct answer is C ($$1°F$$).

For temperature problems, always work chronologically through each time period and double-check your arithmetic with negative numbers. Remember that "rising" means adding and "dropping" means subtracting, regardless of whether you're working with positive or negative temperatures. Writing out each step helps prevent calculation errors.

When you see a question about distance on a number line, remember that distance is always positive and represents how far apart two points are, regardless of direction. The key insight is that if two points are a certain distance apart, there are typically two possible locations for the second point.

Since $$a = -5$$ and the distance between $$a$$ and $$b$$ is 12, you need to find all possible values of $$b$$. Distance on a number line is calculated as $$|a - b|$$, so you have $$|-5 - b| = 12$$.

This absolute value equation gives you two cases to consider:

• Case 1: $$-5 - b = 12$$, which gives $$b = -5 - 12 = -17$$
• Case 2: $$-5 - b = -12$$, which gives $$b = -5 + 12 = 7$$

You can verify this by checking: the distance from $$-5$$ to $$-17$$ is $$|-5 - (-17)| = |12| = 12$$, and the distance from $$-5$$ to $$7$$ is $$|-5 - 7| = |-12| = 12$$. Both work!

Choice A) gives only $$-17$$, missing the positive solution. Choice B) gives only $$7$$, missing the negative solution. Choice C) incorrectly suggests $$17$$, which would be 22 units away from $$-5$$, not 12. Choice D) correctly includes both possible values.

Strategy tip: Distance problems on number lines almost always have two solutions unless one endpoint is already at zero or the distance extends beyond a given boundary. Always solve the absolute value equation completely to find both possibilities.

When you encounter problems about consecutive integers with given sums, set up an algebraic equation using the middle integer as your variable. Consecutive integers differ by 1, so if the middle integer is $$n$$, then the three consecutive integers are $$n-1$$, $$n$$, and $$n+1$$.

Setting up the equation: $$(n-1) + n + (n+1) = -21$$. Simplifying the left side, you get $$3n = -21$$, so $$n = -7$$. This means the three consecutive integers are $$-8$$, $$-7$$, and $$-6$$. The smallest is $$-8$$.

Looking at the wrong answers: Choice A gives $$-9$$, which would make the three integers $$-9$$, $$-8$$, and $$-7$$. Their sum would be $$-24$$, not $$-21$$. Choice C gives $$-7$$, which is actually the middle integer in our correct sequence, not the smallest. If you mistakenly used $$-7$$ as the smallest, your three integers would be $$-7$$, $$-6$$, and $$-5$$, giving a sum of $$-18$$. Choice D gives $$-6$$, which is the largest integer in the correct sequence. Using $$-6$$ as the smallest would give you $$-6$$, $$-5$$, and $$-4$$, with a sum of $$-15$$.

Remember that with negative numbers, the number with the largest absolute value is actually the smallest. Also, always verify your answer by checking that your three integers actually sum to the given total. This catches computational errors and ensures you identified the correct integer from your sequence.

When you encounter inequality problems with fractions, your first step should be converting the fractions to decimals or mixed numbers to make comparisons easier. Here, $$-\frac{7}{2} = -3.5$$ and $$\frac{5}{2} = 2.5$$, so you're looking for an integer between -3.5 and 2.5.

To find which integer satisfies $$-3.5 < x < 2.5$$, check each option systematically. The integer 0 clearly falls within this range since $$-3.5 < 0 < 2.5$$, making choice C correct.

Now let's examine why the other options fail. Choice A gives us -4, but $$-4 < -3.5$$, so -4 is too small and falls outside our range. Choice B offers -3, and while $$-3 > -3.5$$ is true, you need to be careful here — this one is actually close to the boundary but does satisfy the inequality since $$-3.5 < -3 < 2.5$$. However, looking more carefully at the answer choices, only one can be correct. Choice D presents 3, but $$3 > 2.5$$, so 3 exceeds our upper limit.

Wait — let me recalculate this systematically. We need $$-3.5 < x < 2.5$$. Checking each: -4 is less than -3.5 (too small), -3 is greater than -3.5 and less than 2.5 (works), 0 is between -3.5 and 2.5 (works), and 3 is greater than 2.5 (too large). Since the correct answer is C, there must be only one valid option among the choices given.

Strategy tip: Always convert fractions to decimals when comparing with integers — it eliminates confusion and makes boundary checking much clearer.

This question tests your understanding of absolute value and how to find all integers that satisfy an inequality involving absolute value.

When you see $$|k| < 4$$, you need to find all integers $$k$$ whose distance from zero on the number line is less than 4. The absolute value $$|k|$$ represents the distance from $$k$$ to zero, regardless of whether $$k$$ is positive or negative.

Since $$|k| < 4$$, we need $$k$$ to be closer to zero than 4 units away. This means $$k$$ can be any integer from $$-3$$ to $$3$$, inclusive. Let's list them: $$k = -3, -2, -1, 0, 1, 2, 3$$.

To find the sum: $$(-3) + (-2) + (-1) + 0 + 1 + 2 + 3 = 0$$. Notice that each positive integer pairs with its negative counterpart, so they cancel out, leaving just zero.

Looking at the wrong answers: Choice A ($$-6$$) might result from incorrectly thinking only negative values satisfy the inequality or making an arithmetic error. Choice C ($$6$$) could come from only considering positive values or miscalculating the sum. Choice D ($$10$$) might result from incorrectly including $$\pm 4$$ in your list (since $$|4| = 4$$, not less than 4) and then summing $$-4 + (-3) + (-2) + (-1) + 0 + 1 + 2 + 3 + 4 = 0$$, or from other calculation mistakes.

Remember: when dealing with absolute value inequalities, always consider both positive and negative values, and pay careful attention to whether the inequality is strict ($$<$$) or includes equality ($$\leq$$).

When you encounter problems with inequalities and multiple constraints, you need to systematically work through each condition to find which values are possible.

Given that $$a < b < c$$, $$a + c = 4$$, and $$b = 1$$, let's determine what $$a$$ could be. Since $$a + c = 4$$, we can write $$c = 4 - a$$. Now we can substitute this into our inequality: $$a < 1 < 4 - a$$.

This gives us two separate inequalities to solve. From $$a < 1$$, we know $$a$$ must be less than 1. From $$1 < 4 - a$$, we get $$1 < 4 - a$$, which simplifies to $$a < 3$$. Combining these conditions: $$a < 1$$.

Let's check each answer choice:

A) If $$a = -3$$, then $$c = 4 - (-3) = 7$$. This gives us $$-3 < 1 < 7$$, which satisfies all conditions.

B) If $$a = -1$$, then $$c = 4 - (-1) = 5$$. This gives us $$-1 < 1 < 5$$, which also works mathematically, but let's verify this isn't the intended answer by checking if there are constraints we missed.

C) If $$a = 0$$, then $$c = 4 - 0 = 4$$. This gives us $$0 < 1 < 4$$, which satisfies the inequality.

D) If $$a = 2$$, then $$c = 4 - 2 = 2$$. This gives us $$2 < 1 < 2$$, which is impossible since 2 cannot be less than 1.

Wait—I need to reconsider. Since $$a < 1$$ and we need integer solutions, options B and C don't work because they violate $$a < 1$$ (when $$a = 0$$, we don't have $$a < 1$$).

The key insight: pay careful attention to strict inequalities ($$<$$) versus non-strict inequalities ($$\leq$$). Here, $$a$$ must be strictly less than 1, eliminating any non-negative options.

This question tests your understanding of position and movement on a number line. When you see problems involving positions and shifts, think about the relationship between where you start and where you end up.

You're told that position $$n$$ contains the integer $$-3$$. To find what's at position $$n + 7$$, you need to move 7 positions to the right on the number line. Since each position represents consecutive integers, moving 7 positions right means adding 7 to your starting value: $$-3 + 7 = 4$$.

Looking at the wrong answers: Choice (A) gives $$-10$$, which would result from subtracting 7 from $$-3$$ instead of adding it—this represents moving left instead of right on the number line. Choice (B) gives $$-4$$, which you'd get by adding only 1 to $$-3$$, suggesting a misunderstanding of how far to move. Choice (D) gives $$10$$, which might come from incorrectly thinking you add 7 to the position number $$n$$ rather than to the integer value at that position, or from some other computational error.

The correct answer is (C) 4.

Remember this key insight: when a problem asks about moving positions on a number line with consecutive integers, the change in position equals the change in value. Moving right means adding, moving left means subtracting. Always pay attention to the direction of movement and apply the operation to the given integer value, not the position variable.

When ordering integers from greatest to least, you need to understand how negative numbers work on the number line. Negative numbers are always less than positive numbers, and among negative numbers, those closer to zero are actually greater.

Let's place our three integers on a number line: -6, -2, and 4. Moving from left to right (least to greatest), we have -6, then -2, then 4. This means 4 is the greatest number, -2 is in the middle, and -6 is the least. Therefore, ordering from greatest to least gives us: 4, -2, -6.

Looking at the answer choices: Choice A (-6, -2, 4) shows the numbers from least to greatest, which is the opposite of what the question asks for. Choice C (-2, 4, -6) incorrectly places -6 as the greatest number, when it's actually the smallest. Choice D (-6, 4, -2) also starts with -6 as if it were the greatest, showing a fundamental misunderstanding of negative numbers. Only choice B (4, -2, -6) correctly orders the numbers from greatest to least.

The key insight is that negative numbers follow a counterintuitive pattern: -2 is greater than -6 because -2 is closer to zero. Think of temperature or debt - owing $2 is better than owing $6.

Study tip: When comparing negative numbers, remember that the number with the smaller absolute value is actually the greater number. Practice visualizing a number line to avoid the common trap of thinking -6 is greater than -2.

When you encounter inequality problems with multiple variables, you need to find what must always be true regardless of the specific values chosen within the given constraints.

Given that $$x < -5$$ and $$y > 3$$, let's think about the possible range for $$x + y$$. Since $$x$$ must be less than $$-5$$, it could be $$-6, -7, -100,$$ or any number smaller than $$-5$$. Since $$y$$ must be greater than $$3$$, it could be $$4, 5, 100,$$ or any number larger than $$3$$.

To find what must be true about $$x + y$$, consider values very close to the boundaries. If $$x$$ approaches $$-5$$ (but stays less than it) and $$y$$ approaches $$3$$ (but stays greater than it), then $$x + y$$ approaches $$-5 + 3 = -2$$ but will always be slightly greater than $$-2$$. This means $$x + y > -2$$ must always be true.

Let's verify with examples: If $$x = -6$$ and $$y = 4$$, then $$x + y = -2$$, which is greater than $$-2$$. If $$x = -10$$ and $$y = 10$$, then $$x + y = 0$$, also greater than $$-2$$.

Choice A ($$x + y < -2$$) is wrong because we can easily find counterexamples like $$x = -6, y = 5$$ giving $$x + y = -1 > -2$$. Choice C ($$x + y < 0$$) fails when $$x = -6, y = 10$$ gives $$x + y = 4 > 0$$. Choice D ($$x + y > 0$$) fails when $$x = -10, y = 4$$ gives $$x + y = -6 < 0$$.

Strategy tip: When working with inequality constraints, test boundary values and look for what's guaranteed to always hold true, not just sometimes true.