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Absolute Value

8 questions

All types8Multiple choice4Numeric2Fill in the blank2
Question 1Question →

If x=−3x = -3, what is the value of the expression 8−3∣2x−7∣8 - 3|2x - 7|?

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Answer: -31

Answer

-31
The expression is evaluated using the correct order of operations. First, the value inside the absolute value is simplified by substituting -3, yielding -13. The absolute value of -13 is positive 13. Finally, multiplying 13 by 3 gives 39, and subtracting 39 from 8 results in -31.

Step-by-Step Solution

1
Substitute -3 for x in the expression.
8−3∣2(−3)−7∣8 - 3|2(-3) - 7|
To evaluate the expression for the specific variable value.
2
Multiply the numbers inside the absolute value.
8−3∣−6−7∣8 - 3|-6 - 7|
Following the order of operations, calculations grouped inside absolute value bars must be simplified first.
3
Subtract the remaining numbers inside the absolute value.
8−3∣−13∣8 - 3|-13|
Continuing to simplify the contents within the absolute value grouping.
4
Evaluate the absolute value of -13.
8−3(13)8 - 3(13)
The absolute value of any non-zero number is its positive distance from zero.
5
Multiply 3 by 13.
8−398 - 39
According to the order of operations (PEMDAS), multiplication must be performed before subtraction.
6
Subtract 39 from 8.
-31
This is the final operation to find the total value of the expression.

Key Concept

Evaluating absolute value expressions with order of operations
Question 2Question →

If x=−3x = -3, y=−5y = -5, and z=2z = 2, what is the value of the expression −2∣x2−∣y−3z∣∣+4x-2|x^2 - |y - 3z|| + 4x?

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Answer: -16

Answer

-16
The correct answer is found by strictly adhering to the order of operations and properties of absolute values. First, the inner absolute value |-5 - 3(2)| evaluates to |-11|, which is 11. The expression then simplifies to -2|(-3)² - 11| + 4(-3). Squaring -3 gives 9, so the expression inside the outer absolute value becomes 9 - 11, which is -2. The absolute value of -2 is 2. Multiplying this by the leading coefficient of -2 gives -4, and adding 4(-3), which is -12, results in -16.

Step-by-Step Solution

1
Substitute the given variable values into the expression.
-2|(-3)² - |-5 - 3(2)|| + 4(-3)
To prepare the expression for evaluation by replacing variables with their known numeric values.
2
Evaluate the innermost operations following the order of operations, starting with the inner absolute value.
-5 - 3(2) = -5 - 6 = -11, so |-11| = 11
The order of operations (PEMDAS) requires evaluating multiplication before subtraction, and operations inside the innermost grouping symbols first.
3
Evaluate the exponent and substitute the result of the inner absolute value back into the expression.
-2|9 - 11| - 12
Exponents must be evaluated before subtraction within the remaining absolute value.
4
Evaluate the subtraction inside the remaining absolute value and apply the absolute value function.
|9 - 11| = |-2| = 2
The absolute value of any non-zero integer is its positive counterpart.
5
Multiply by the leading coefficient and add the final term.
-2(2) - 12 = -4 - 12 = -16
Multiplication must be performed before subtraction to find the final value.

Key Concept

Evaluating multi-step algebraic expressions with nested absolute values
Question 3Question →

Determine the solutions for xx in the given absolute value equation.

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Given the equation 2∣5−x∣−4=102|5 - x| - 4 = 10, the smaller value of xx is and the larger value of xx is .
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Answer

The smaller value is -2 and the larger value is 12.
To solve 2∣5−x∣−4=102|5 - x| - 4 = 10, first isolate the absolute value term. Adding 4 to both sides yields 2∣5−x∣=142|5 - x| = 14. Dividing both sides by 2 gives ∣5−x∣=7|5 - x| = 7. The absolute value of an expression equals a positive number when the expression itself equals that positive number or its negative counterpart. Therefore, we set up two cases: 5−x=75 - x = 7 and 5−x=−75 - x = -7. Solving the first case gives −x=2-x = 2, so x=−2x = -2. Solving the second case gives −x=−12-x = -12, so x=12x = 12. Comparing the two solutions, the smaller value is −2-2 and the larger value is 1212.

Step-by-Step Solution

1
Isolate the absolute value expression.
Add 4 to both sides to get 2∣5−x∣=142|5 - x| = 14, then divide by 2 to get ∣5−x∣=7|5 - x| = 7.
The absolute value must be isolated before separating the equation into cases.
2
Set up the positive case equation.
5−x=75 - x = 7
The first case assumes the expression inside the absolute value is positive or zero.
3
Solve the positive case.
Subtract 5 to get −x=2-x = 2, which means x=−2x = -2.
This finds the first potential value for xx.
4
Set up the negative case equation.
5−x=−75 - x = -7
The second case accounts for the expression inside the absolute value being negative.
5
Solve the negative case.
Subtract 5 to get −x=−12-x = -12, which means x=12x = 12.
This finds the second potential value for xx.

Key Concept

Solving Linear Absolute Value Equations
Estimated Time:1m 30s
Question 4Question →

What is the value of the expression 12−2∣−4+1∣12 - 2|-4 + 1|?

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Answer: 6

Answer

6
The expression is evaluated by first simplifying the terms inside the absolute value to get |-3|, which equals positive 3. Multiplying this by the coefficient 2 gives 6. Finally, subtracting 6 from 12 yields the correct answer of 6.

Step-by-Step Solution

1
Simplify the expression inside the absolute value bars.
−4+1=−3-4 + 1 = -3
According to the order of operations, operations inside absolute value bars (which act as grouping symbols) must be performed first.
2
Evaluate the absolute value of the result.
∣−3∣=3|-3| = 3
The absolute value of a number represents its distance from zero, which is always positive.
3
Multiply the result by the coefficient outside the absolute value.
2×3=62 \times 3 = 6
Multiplication must be performed before subtraction according to the order of operations (PEMDAS).
4
Subtract the product from the initial value.
12−6=612 - 6 = 6
This completes the evaluation of the expression to find the final numerical value.

Key Concept

Absolute Value and Order of Operations
Question 5Question →

What is the value of the following expression?

∣−4−∣−9∣∣|-4 - |-9||

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Answer: 13

Answer

13
The correct answer is found by first evaluating the innermost absolute value, ∣−9∣=9|-9| = 9. Substituting this value back into the expression gives ∣−4−9∣|-4 - 9|. Simplifying the subtraction inside yields ∣−13∣|-13|. Finally, the absolute value of −13-13 is 1313.

Step-by-Step Solution

1
Evaluate the inner absolute value.
∣−9∣=9|-9| = 9
According to the order of operations, we must simplify the expression from the inside out. The absolute value of a negative number is its positive counterpart.
2
Substitute and simplify the remaining expression.
∣−4−9∣=∣−13∣|-4 - 9| = |-13|
Replacing ∣−9∣|-9| with 99 gives the subtraction −4−9-4 - 9, which simplifies to −13-13.
3
Evaluate the outer absolute value.
∣−13∣=13|-13| = 13
The absolute value represents the distance from zero on the number line, which is always non-negative.

Key Concept

Absolute Value
Question 6Question →

Given x=−4x = -4 and y=3y = 3, what is the exact value of the expression 3∣x−y∣−∣y2−x2∣3|x - y| - |y^2 - x^2|?

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Answer: 14

Answer

14
By correctly substituting the variables, following the order of operations, and applying the definition of absolute value, the expression simplifies to 3(7)−73(7) - 7, which evaluates precisely to 1414.

Step-by-Step Solution

1
Substitute the given values for xx and yy into the expression.
3∣−4−3∣−∣(3)2−(−4)2∣3|-4 - 3| - |(3)^2 - (-4)^2|
Variable substitution is the first step in evaluating an algebraic expression.
2
Simplify the expression inside the first absolute value bars.
∣−4−3∣=∣−7∣|-4 - 3| = |-7|
Absolute value bars act as grouping symbols, so operations inside them must be performed first.
3
Evaluate the exponents inside the second absolute value bars.
∣(3)2−(−4)2∣=∣9−16∣|(3)^2 - (-4)^2| = |9 - 16|
Following the order of operations, exponents are evaluated before subtraction.
4
Simplify the expression inside the second absolute value bars.
∣9−16∣=∣−7∣|9 - 16| = |-7|
Complete the subtraction inside the grouping symbols.
5
Apply the absolute value function to both simplified terms.
3(7)−73(7) - 7
The absolute value of any non-zero real number is its positive counterpart: ∣−7∣=7|-7| = 7.
6
Perform the final multiplication and subtraction.
21−7=1421 - 7 = 14
Multiplication is performed before subtraction according to the standard order of operations.

Key Concept

Evaluating algebraic expressions involving absolute value, exponents, and signed numbers.
Estimated Time:1m 30s
Question 7Question →

If y=−3y = -3, which of the following represents all possible values for xx in the equation −2−2∣x−y∣=−16-2 - 2|x - y| = -16?

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Answer: 44 and −10-10

Answer

The possible values for xx are 44 and −10-10.
The correct answer accurately substitutes the negative value, isolates the absolute value term to yield ∣x+3∣=7|x + 3| = 7, and evaluates both the positive and negative branches to find the two valid solutions for xx.

Step-by-Step Solution

1
Substitute the value y=−3y = -3 into the original equation.
−2−2∣x−(−3)∣=−16⇒−2−2∣x+3∣=−16-2 - 2|x - (-3)| = -16 \Rightarrow -2 - 2|x + 3| = -16
To reduce the equation to a single variable so it can be solved.
2
Begin isolating the absolute value expression by adding 22 to both sides.
−2∣x+3∣=−16+2⇒−2∣x+3∣=−14-2|x + 3| = -16 + 2 \Rightarrow -2|x + 3| = -14
To undo the constant subtraction outside the absolute value, following reverse order of operations.
3
Divide both sides of the equation by −2-2.
∣x+3∣=7|x + 3| = 7
To completely isolate the absolute value expression.
4
Set up two separate equations for the positive and negative pathways.
x+3=7x + 3 = 7 and x+3=−7x + 3 = -7
The expression inside the absolute value can evaluate to either 77 or −7-7 and still have an absolute distance of 77 from zero.
5
Solve both resulting linear equations for xx.
x=4x = 4 and x=−10x = -10
To find all possible values that satisfy the original equation.

Key Concept

Solving linear absolute value equations requires accurately substituting variables, strictly following the order of operations to isolate the absolute value, and creating two distinct pathways (positive and negative) to find all solutions.
Estimated Time:1m 30s
Question 8Question →

Evaluate the following mathematical expression and determine the final integer result.

Fill in the blanks below

The value of the expression ∣−15+3∣2−∣−4×2∣\frac{|-15 + 3|}{2} - |-4 \times 2| is .
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Answer

The final evaluated result is -2.
The correct result is found by first evaluating the expressions inside the absolute value symbols, which act as grouping symbols. The expression ∣−15+3∣|-15 + 3| becomes ∣−12∣|-12|, which evaluates to 1212. The second part, ∣−4×2∣|-4 \times 2|, becomes ∣−8∣|-8|, which evaluates to 88. Substituting these simplified values back into the original expression yields 122−8\frac{12}{2} - 8. Following the order of operations, division is performed first, giving 66. Finally, 6−86 - 8 results in −2-2.

Step-by-Step Solution

1
Evaluate the operations inside the first absolute value bars.
∣−15+3∣=∣−12∣|-15 + 3| = |-12|
Order of operations requires treating absolute value bars as grouping symbols, meaning the arithmetic inside must be completed first.
2
Apply the absolute value to the first result.
∣−12∣=12|-12| = 12
The absolute value of a number represents its distance from zero, which is always positive.
3
Evaluate the operations inside the second absolute value bars.
∣−4×2∣=∣−8∣=8|-4 \times 2| = |-8| = 8
Perform the multiplication inside the grouping symbol first, then take the absolute value of the negative product.
4
Substitute the evaluated absolute values back into the expression and solve.
122−8=6−8=−2\frac{12}{2} - 8 = 6 - 8 = -2
Perform division before subtraction according to the standard order of operations (PEMDAS).

Key Concept

Absolute Value and Order of Operations
Absolute Value Practice Questions — Armed Services Vocational Aptitude Battery (ASVAB) | Examkin