Definition of absolute value and simplifying expressions

Key idea

Absolute value can be defined piecewise: \[ \lvert x\rvert = \begin{cases} x, & x \ge 0 \\ -x, & x < 0 \end{cases} \] This definition explains why \(\lvert x\rvert\) is always nonnegative. A common trap is mixing up \(\lvert -x\rvert\) and \(-\lvert x\rvert\): \(\lvert -x\rvert=\lvert x\rvert\), but \(-\lvert x\rvert\) is the negative of a nonnegative number.

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Summary

• \(\lvert x\rvert\) is always nonnegative and can be defined piecewise.
• Be careful with a negative sign outside the bars: \(-\lvert x\rvert\) is negative (unless \(x=0\)).

Absolute value as distance on the number line

Key idea

\(\lvert x\rvert\) is the distance from \(0\) to \(x\). More generally, the distance between two numbers \(a\) and \(b\) is: \[ \text{distance}(a,b)=\lvert a-b\rvert. \] This is why \(\lvert a-b\rvert=\lvert b-a\rvert\): distance does not depend on direction.

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Summary

• \(\lvert x\rvert\) is distance from \(0\).
• \(\lvert a-b\rvert\) is distance between \(a\) and \(b\).

Solving absolute value equations

Key idea

If \(c \ge 0\), then: \[ \lvert A\rvert=c \quad \Rightarrow \quad A=c \text{ or } A=-c. \] If \(c < 0\), there is no solution, because an absolute value cannot be negative.

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Summary

• For \(\lvert A\rvert=c\) with \(c\ge 0\), solve \(A=c\) and \(A=-c\).
• If \(c<0\), there are no solutions.

Absolute value inequalities: less than and at most

Key idea

When \(c \ge 0\): \[ \lvert A\rvert < c \;\Rightarrow\; -c < A < c, \qquad \lvert A\rvert \le c \;\Rightarrow\; -c \le A \le c. \] This matches the distance meaning: \(\lvert A\rvert < c\) means “\(A\) is within \(c\) units of \(0\)”.

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Summary

• \(\lvert A\rvert<c\) becomes \(-c<A<c\) (one “between” inequality).
• \(\lvert A\rvert\le c\) becomes \(-c\le A\le c\).

Absolute value inequalities: greater than and at least

Key idea

When \(c \ge 0\): \[ \lvert A\rvert > c \;\Rightarrow\; A > c \text{ or } A < -c, \qquad \lvert A\rvert \ge c \;\Rightarrow\; A \ge c \text{ or } A \le -c. \] This matches distance: \(\lvert A\rvert>c\) means “\(A\) is more than \(c\) units away from \(0\)”, so you get two separated regions.

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Summary

• \(\lvert A\rvert>c\) becomes \(A>c\) or \(A<-c\) (a union of two intervals).
• \(\lvert A\rvert\ge c\) becomes \(A\ge c\) or \(A\le -c\).

Graphs of absolute value and the piecewise function view

Key idea

The graph of \(y=\lvert x\rvert\) is a “V” with vertex at \((0,0)\). You can view it as two lines: \[ y=\lvert x\rvert = \begin{cases} x, & x \ge 0 \\ -x, & x < 0 \end{cases} \] A useful transformation pattern is: \[ y=\lvert x-h\rvert + k, \] which shifts the vertex to \((h,k)\).

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Summary

• \(y=\lvert x\rvert\) is a V-shape with vertex at \((0,0)\).
• \(y=\lvert x-h\rvert+k\) shifts the vertex to \((h,k)\).
• The piecewise definition explains the shape: one line for \(x\ge 0\), another for \(x<0\).

Why absolute value matters

Where absolute value shows up

• Distance and geometry: how far apart values are on a line (\(\lvert a-b\rvert\)).
• Error and tolerance: “within” a target value (\(\lvert x-\text{target}\rvert \le \text{tolerance}\)).
• Algebra and graphs: V-shaped functions and transformations.
• Inequalities: turning “within” or “at least this far” statements into intervals.

Worked example: tolerance inequality

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Final recap

• \(\lvert x\rvert\) is distance and is always nonnegative.
• Simplify carefully: \(\lvert -x\rvert=\lvert x\rvert\), but \(-\lvert x\rvert\) is negative (unless \(x=0\)).
• Equation: \(\lvert A\rvert=c \Rightarrow A=c\) or \(A=-c\) (for \(c\ge 0\)).
• Inequality (less than): \(\lvert A\rvert<c \Rightarrow -c<A<c\).
• Inequality (greater than): \(\lvert A\rvert>c \Rightarrow A>c\) or \(A<-c\).
• Graph: \(y=\lvert x-h\rvert+k\) has vertex \((h,k)\).