Absolute Value and Order - Quantitative Reasoning

Identify the correct order from least to greatest: $|-1|,\ -\frac{3}{2},\ \frac{4}{3}$ .

$-\frac{3}{2}<|-1|<\frac{4}{3}$ . Ordering starts with negative -1.5, then 1 from | -1 |, then positive 1.333.
What is the solution set of the inequality $|2x+1|\le 5$ ?

$-3\le x\le 2$ . Solving |2x+1|<=5 gives -5 <= 2x+1 <=5, subtract 1 then divide by 2.
Which is larger: $-\frac{3}{2}$ or $\left| -\frac{5}{4} \right|$ ?

$\left| -\frac{5}{4} \right|=\frac{5}{4}$ is larger. Absolute value turns -5/4 positive to 1.25, which is greater than -1.5.
What is the smallest possible value of $|x|$ for real $x$ , and when does it occur?

Minimum is $0$ , occurring at $x=0$ . The absolute value |x| is always non-negative, reaching its minimum of 0 only when x=0.
What is the solution set of the inequality $|x-3|<2$ ?

$1<x<5$ . Solving |x-3|<2 expands to -2 < x-3 < 2, adding 3 to all parts.
Which is larger: $|2-9|$ or $|2-6|$ ?

$|2-9|=7$ is larger. Distance from 2 to 9 is 7, exceeding distance to 6 which is 4.
What inequality is equivalent to $|x|\le a$ for $a>0$ ?

$-a\le x\le a$ . For positive a, |x|<=a includes all x within or at distance a from 0.
What is the definition of absolute value $|x|$ written as a piecewise function?

$|x|=\begin{cases}x,&x\ge 0\\-x,&x<0\end{cases}$ . The absolute value function returns the non-negative value by taking x when x is non-negative and its negation when x is negative.
Which is larger: $|-7|$ or $|-5|$ ?

$|-7|$ is larger. Larger magnitude negative input to absolute value yields larger output, as 7>5.
What is the solution set of the inequality $|x+2|\ge 4$ ?

$x\le -6$ or $x\ge 2$ . Solving |x+2|>=4 splits into x+2 <= -4 or x+2 >=4, subtracting 2.
What is the solution set of $|x|>0$ ?

$x\ne 0$ . |x|>0 excludes only x=0 where |x|=0, including all other reals.
What is the solution set of $|x|<0$ ?

No solution; empty set. Since |x| is always >=0, it cannot be less than 0, yielding no solutions.
Which is larger: $|x-1|$ or $|x-4|$ when $x=0$ ?

$|0-4|$ is larger. At x=0, distance to 4 is 4, greater than distance to 1 which is 1.
What is the distance interpretation of $|a-b|$ on the number line?

$|a-b|$ is the distance between $a$ and $b$ . On the number line, the absolute difference |a-b| measures the positive distance separating points a and b.
What inequality is equivalent to $|x|\ge a$ for $a>0$ ?

$x\le -a$ or $x\ge a$ . For positive a, |x|>=a includes all x at or beyond distance a from 0.
Identify the correct order from least to greatest: $-2,\ | -3 |,\ 1$ .

$-2<1<|-3|$ . Ordering -2 (negative), then 1 (positive), then 3 from | -3 |.
What is the solution set of $|x|\ge a$ when $a<0$ ?

All real numbers. For negative a, |x|>=0 is always greater than or equal to a, so all reals satisfy.
What is the ordering rule comparing $|a|$ and $|b|$ using squares?

$|a|<|b|\iff a^2<b^2$ . Squaring both sides preserves the inequality since squares equal the squares of absolutes and are monotonic for non-negative values.
What is the solution set of $|x|<a$ when $a<0$ ?

No solution; empty set. For negative a, |x|>=0 cannot be less than a negative number, so no solutions.
Which is larger: $-3$ or $|-4|$ ?

$|-4|=4$ is larger. Absolute value makes |-4| positive 4, which exceeds -3 on the number line.
What is the largest possible value of $|x|$ when $x$ ranges over all real numbers?

No largest value; $|x|$ is unbounded above. Since x can be arbitrarily large positive or negative, |x| can exceed any bound with no maximum.
What identity relates absolute value to a square root for real $x$ ?

$|x|=\sqrt{x^2}$ . The square root of x squared yields the non-negative root, matching the absolute value.
What inequality is equivalent to $|x|>a$ for $a>0$ ?

$x<-a$ or $x>a$ . For positive a, |x|>a means x is more than distance a from 0, either left of -a or right of a.
What is the solution set of $|x|\le 0$ ?

$x=0$ . Since |x|>=0 and equals 0 only at x=0, |x|<=0 holds solely there.

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