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Lesson Worksheet: Solving Absolute Value Problems Mathematics

In this worksheet, we will practice solving equations involving absolute values algebraically and graphically.

Q1:

Given the function (𝑥)=5|𝑥1|3, 𝑥, solve the equation (𝑥)=14𝑥+2.

  • A𝑥=421 or 𝑥=0
  • B𝑥=0 or 𝑥=10
  • C𝑥=0 or 𝑥=4019
  • D𝑥=0 or 𝑥=1940
  • E𝑥=421 or 𝑥=4019

Q2:

Which of the following graphs represents 𝑓(𝑥)=|8+𝑥|3?

  • A
  • B
  • C
  • D
  • E

Find its domain and range.

  • AThe domain is , and the range is [8,).
  • BThe domain is [3,), and the range is .
  • CThe domain is , and the range is (3,).
  • DThe domain is , and the range is [3,).
  • EThe domain is , and the range is (8,).

Q3:

Which of the following functions is represented by the given graph?

  • A𝑓(𝑥)=3|2𝑥|
  • B𝑓(𝑥)=3+|2+𝑥|
  • C𝑓(𝑥)=3+|2𝑥|
  • D𝑓(𝑥)=3+|2𝑥|
  • E𝑓(𝑥)=3|2𝑥|

Find its domain and range.

  • AThe domain is , and the range is [3,).
  • BThe domain is , and the range is (,2].
  • CThe domain is , and the range is [0,).
  • DThe domain is (,3], and the range is .
  • EThe domain is , and the range is (,3].

Q4:

Find the range of the function 𝑓(𝑥)=|2𝑥2|.

  • A{1}
  • B{1}
  • C
  • D[1,)
  • E[0,)

Q5:

Find the domain and range of the function 𝑓(𝑥)=4|𝑥5|1.

  • AThe domain is (,1] and the range is .
  • BThe domain is {5} and the range is {1}.
  • CThe domain is and the range is (,1).
  • DThe domain is and the range is (,1].

Q6:

The graph in figure (i) is of 𝑓(𝑥)=52|𝑥|+12𝑥, which could also be written as follows 𝑓(𝑥)=3𝑥,𝑥02𝑥,𝑥<0.

Find the values of 𝑎 and 𝑏 that would make graph (ii) that of 𝑔(𝑥)=𝑎|𝑥5|+𝑏(𝑥5).

  • A𝑎=3,𝑏=1
  • B𝑎=1,𝑏=5
  • C𝑎=2,𝑏=1
  • D𝑎=3,𝑏=1
  • E𝑎=2,𝑏=1

Q7:

The equations 𝑝(𝑥) and 𝑞(𝑥) are defined by 𝑝(𝑥)=8𝑥,𝑥,𝑞(𝑥)=2|𝑥+𝑎|5,𝑥, where 𝑎 is a constant.

The equation 𝑝(𝑥)=𝑞(𝑥) has exactly one real root. Find the value of 𝑎.

Q8:

A function 𝑓 is defined by 𝑓𝑥62|𝑥+2|. A sketch of the graph of this function is shown below.

Why does 𝑓not exist?

  • A𝑓(𝑥) is a one-to-one function.
  • B𝑓(𝑥) is not a straight line.
  • C𝑓(𝑥) is a many-to-one function.
  • D𝑓(𝑥) is a one-to-many function.
  • E𝑓(𝑥) is not defined for all values of 𝑥.

Solve the inequality 𝑓(𝑥)>3.

  • A72<𝑥<0
  • B12<𝑥<72
  • C72<𝑥<12
  • D72<𝑥<12
  • E12<𝑥<0

Q9:

The function 𝑔 is defined as 𝑔(𝑥)=3|𝑥+3|6, 𝑥. Which of the following sketches is correctly shaded to represent the region that satisfies 𝑦𝑔(𝑥)?

  • A
  • B
  • C
  • D
  • E

This lesson includes 70 additional question variations for subscribers.

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