Number lines, absolute value and estimation

01 / UNDERSTAND & APPLY

Order describes position; distance describes separation

The real number line increases from left to right. Negative numbers farther from zero are smaller: −8<−3. Integers, fractions and irrational numbers all occupy positions on the same line. A number is not positive merely because its absolute value is large.

GRE number lines and coordinate/data graphs are drawn to scale; ordinary geometric figures need not be. Read the marked units and tick spacing rather than imposing a scale from a different figure. A point between two labeled ticks is not necessarily an integer.

02 / UNDERSTAND & APPLY

Absolute value removes direction, not information about location

|x| is the distance from x to zero, so it is never negative. It equals x for x≥0 and −x for x<0. The expression |x−a| is the distance between x and a. Distance between a and b can be written |a−b| or |b−a|.

Absolute value does not distribute over addition: |−4+7|=3, while |−4|+|7|=11. The latter sum is always at least as large as the absolute value of the sum. For a product, |ab|=|a||b|.

03 / UNDERSTAND & APPLY

An equation usually identifies two locations

For c>0, |x−a|=c means x=a−c or x=a+c. For c=0, the only solution is x=a. For c<0, there are no real solutions because a distance cannot be negative. Isolate the absolute value before splitting into cases.

Check both solutions in the original equation, especially when other expressions or denominators are present. A condition such as x>0 may eliminate one location.

04 / UNDERSTAND & APPLY

Distance inequalities describe intervals or two rays

For r>0, |x−a|<r means a−r<x<a+r: the point lies within r units of a. The condition |x−a|>r means x<a−r or x>a+r: the point is outside. Replace strict symbols with inclusive ones when boundary distance r is allowed.

An open circle excludes an endpoint; a closed circle includes it. In interval notation, parentheses exclude and brackets include endpoints. Infinity is never a finite included endpoint. Distinguish “and” for an interior interval from “or” for separated exterior rays.

05 / UNDERSTAND & APPLY

When two distances appear, use the breakpoints

To remove absolute values, split the line at points where their inside expressions change sign. Within each region the sign is known, so the formula becomes linear. A formula derived for a region is valid only there; reject algebraic candidates outside it.

For |x−2|+|x−8|, any x from 2 through 8 has total distance 6. Outside that interval the sum grows. The geometric explanation is that inside the segment the two distances partition its length; outside it, the extra distance is traversed twice.

06 / UNDERSTAND & APPLY

Estimation needs a direction and an error tolerance

An estimate is useful only at a precision suited to the question. To compare products of positive numbers, lower and upper bounds can establish a result without exact multiplication. If one quantity is certainly below 30 and another certainly above 30, the ordering is settled.

Rounding each factor independently does not automatically give a rigorous upper bound unless each replacement is upward and the quantities are nonnegative. When signs or reciprocals appear, revisit how inequalities change. Exact equality cannot be justified by equal rounded displays.

07 / UNDERSTAND & APPLY

Use valid cases when location is not fixed

Knowing |x| determines magnitude but often not sign. If |x|=4, x may be −4 or 4. A question involving x itself may therefore be indeterminate even when a question involving x² is fixed.

For QC, list boundary, zero, negative and fractional possibilities permitted by the prompt. A pair of different allowed relationships proves D. A single successful trial cannot prove a universal relationship.

08 / UNDERSTAND & APPLY

Choose a method and answer the requested question

All examples and practice in this lesson are original. For quantitative comparison (QC), use A if Quantity A is greater, B if Quantity B is greater, C if they are equal, and D if the relationship cannot be determined from the information given. A, B or C must hold for every permitted case. To establish D, give two allowed cases producing different relationships; an inability to calculate is not itself evidence for D.

A single-answer question requires one choice. For multiple-answer questions, obey the stated selection count or select every correct option if instructed. Test each option separately; do not stop after finding one. Multiple-answer questions receive credit only when the complete correct set is selected, with no incorrect choices. Numeric entry requires the requested value, with units handled in the calculation; round only when the question directs you to. A single numeric-entry box accepts an integer or decimal; a pair of fraction boxes accepts an integer numerator and a nonzero integer denominator. Fractions need not be reduced, but reducing makes checks easier. Do not enter a percent sign when the requested number is a percentage. Preserve exact fractions until the last step.

A calculator can check arithmetic but cannot decide the domain, interpretation or denominator for you. Estimate sign and size first, enter parentheses carefully, and compare the result with the original conditions. Cancel factors or exploit structure when doing so is faster and more reliable than decimal entry.

09 / UNDERSTAND & APPLY

Original practice: endpoints and signs

Use the QC choices defined earlier.