Modulus & Inequalties
Absolute value (modulus) ∣x∣, and solving modulus equations and inequalities such as ∣f(x)∣=x−3 and g(x)≥f(x) algebraically or graphically, including changing the inequality direction when multiplying by a negative number.
Absolute value (modulus)
The absolute value of x is defined as
This has the effect of making any negative argument positive, and has no impact on positive values:
The absolute value is also known as the modulus.
Solving modulus equations and inequalities
Equations and inequalities with absolute values can show up on IB exams. For example:
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
Inequalities of the form
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:
Nice work completing Modulus & Inequalties, here's a quick recap of what we covered:
Skills covered
Exercises checked off