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Modulus & Inequalties
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Absolute value, inequalities of functions
Want a deeper conceptual understanding? Try our interactive lesson! (Plus Only)
Absolute value (modulus)
AHL 2.16
The absolute value of x is defined as
∣x∣={x,x≥0−x,x<0
This has the effect of making any negative argument positive, and has no impact on positive values:
∣−4∣=4,∣2∣=2
The absolute value is also known as the modulus.
Solving modulus equations and inequalities
AHL 2.16
Equations and inequalities with absolute values can show up on IB exams. For example:
∣2x−4∣<∣x+3∣−2
We recommend solving these graphically, recalling that ∣f(x)∣ has the effect of reflecting vertically (in the x-axis) any part of the graph which is negative.
From the animation, we see the solution is 1<x<5.
For more complicated functions, you can plot the absolute values on your calculator, find the intersections there, and inspect visually where one function is greater than the other.
Solving inequalities of functions
AHL 2.15
Inequalities of the form
g(x)≥f(x)
can be solved either algebraically or with technology.
It is crucial to remember that when multiplying both sides of an inequality by a negative number, the inequality changes direction:
−x<1⇒x>−1.
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