Evaluating Expressions • Absolute Value • Integer Operations
Order of Operations (PEMDAS / GEMDAS)
Work from left to right, following this priority:
1. Grouping – parentheses ( ), brackets [ ], and absolute value bars | |
2. Exponents
3. Multiply & Divide (left to right)
4. Add & Subtract (left to right)
Absolute value bars act as grouping symbols — evaluate the expression inside first.
Absolute Value
The absolute value of a number is its distance from zero on the number line. It is always non-negative.
\( |a| = a \) if \( a \geq 0 \) \( |a| = -a \) if \( a < 0 \)
Examples: \( |-7| = 7 \) \( |0| = 0 \) \( |4 – 9| = |-5| = 5 \)
Integer Operation Rules
Multiplication / Division Same signs → positive result Different signs → negative result
Examples: \( (-3)(-7) = 21 \) \( (-4)(5) = -20 \) \( 12 \div (-3) = -4 \)
Subtraction: Subtracting a negative is the same as adding the positive. \( 4 – (-5) = 4 + 5 = 9 \)
Tip: Always evaluate absolute values and any expressions inside grouping symbols first. Then apply multiplication and division before addition and subtraction.
Evaluate each expression. Show all steps carefully, paying special attention to absolute value and the order of operations.
Evaluate the expression \( -3(-7) + 5 \cdot |-2| \).
First evaluate the absolute value:
$$|-2| = 2$$
Multiply the integers (negative × negative = positive):
$$-3(-7) = 21$$
Multiply:
$$5 \cdot 2 = 10$$
Add:
$$21 + 10 = 31$$
Answer: 31
Evaluate the expression \( 8 – |-5| \cdot 2 \).
Absolute value first:
$$|-5| = 5$$
Multiply:
$$5 \cdot 2 = 10$$
Subtract:
$$8 – 10 = -2$$
Answer: −2
Evaluate the expression \( (-4)(-6) + |-9| \).
Multiply the two negatives:
$$(-4)(-6) = 24$$
Absolute value:
$$|-9| = 9$$
Add:
$$24 + 9 = 33$$
Answer: 33
Evaluate the expression \( |-12| \div 3 – (-5) \).
Absolute value:
$$|-12| = 12$$
Divide:
$$12 \div 3 = 4$$
Subtracting a negative is the same as adding:
$$4 – (-5) = 4 + 5 = 9$$
Answer: 9
Evaluate the expression \( 2 \cdot |-3| – (-8) \).
Absolute value:
$$|-3| = 3$$
Multiply:
$$2 \cdot 3 = 6$$
Subtract the negative:
$$6 – (-8) = 6 + 8 = 14$$
Answer: 14
Evaluate the expression \(-5 + 3 \cdot \left| -4 - 2 \right|\).
Evaluate inside the absolute value first:
$$-4 – 2 = -6$$
$$|-6| = 6$$
Multiply:
$$3 \cdot 6 = 18$$
Add:
$$-5 + 18 = 13$$
Answer: 13
Evaluate the expression \(\left| -8 \right| - 4\left( -3 + \left| -1 \right| \right)\).
Innermost absolute value:
$$|-1| = 1$$
Inside the parentheses:
$$-3 + 1 = -2$$
Multiply:
$$4 \cdot (-2) = -8$$
Absolute value of −8:
$$|-8| = 8$$
Subtract:
$$8 – (-8) = 8 + 8 = 16$$
Answer: 16
Evaluate the expression \((-2)\left[ 6 - \left| -5 \right| \right] + 7\).
Absolute value:
$$|-5| = 5$$
Inside the brackets:
$$6 – 5 = 1$$
Multiply by −2:
$$(-2) \cdot 1 = -2$$
Add 7:
$$-2 + 7 = 5$$
Answer: 5
Evaluate the expression \(15 \div \left| -3 \right| - (-4) \cdot 2\).
Absolute value:
$$|-3| = 3$$
Divide:
$$15 \div 3 = 5$$
Multiply:
$$(-4) \cdot 2 = -8$$
Subtract the negative:
$$5 – (-8) = 5 + 8 = 13$$
Answer: 13
Evaluate the expression \( |-7 + 3| \cdot (-5) + 10 \).
Inside the absolute value:
$$-7 + 3 = -4$$
$$|-4| = 4$$
Multiply:
$$4 \cdot (-5) = -20$$
Add 10:
$$-20 + 10 = -10$$
Answer: −10
Evaluate the expression \(\left| -2 - 3(-4) \right| + \left| -5 \right| \cdot (-1)\).
Innermost multiplication:
$$3(-4) = -12$$
Inside the first absolute value:
$$-2 – (-12) = -2 + 12 = 10$$
$$|10| = 10$$
Second absolute value and multiply:
$$|-5| = 5 \quad \text{and} \quad 5 \cdot (-1) = -5$$
Add:
$$10 + (-5) = 5$$
Answer: 5
Evaluate the expression \(6 - \left| -3 - 5 \right| \cdot (-2) + \left| -1 \right|\).
Inside the first absolute value:
$$-3 – 5 = -8$$
$$|-8| = 8$$
Multiply:
$$8 \cdot (-2) = -16$$
Absolute value of −1:
$$|-1| = 1$$
Now the full expression (left to right for addition/subtraction):
$$6 – (-16) + 1 = 6 + 16 + 1 = 23$$
Answer: 23
Evaluate the expression \((-1)\left[ \left| -8 \right| - 3 \cdot \left| -2 \right| \right] + 4\).
Absolute values:
$$|-8| = 8 \quad \text{and} \quad |-2| = 2$$
Multiply:
$$3 \cdot 2 = 6$$
Inside the brackets:
$$8 – 6 = 2$$
Multiply by −1:
$$(-1) \cdot 2 = -2$$
Add 4:
$$-2 + 4 = 2$$
Answer: 2
Evaluate the expression \(\left| 4 - \left| -9 \right| \right| - 2\left( -3 + \left| -5 \right| \right)\).
Innermost absolute value:
$$|-9| = 9$$
Outer absolute value:
$$4 – 9 = -5 \quad \Rightarrow \quad |-5| = 5$$
Inside the parentheses:
$$|-5| = 5 \quad \Rightarrow \quad -3 + 5 = 2$$
Multiply:
$$2 \cdot 2 = 4$$
Subtract:
$$5 – 4 = 1$$
Answer: 1
Evaluate the expression \(-\left| -7 + 2 \cdot \left| -3 \right| \right| \cdot 4 + 5\).
Innermost absolute value:
$$|-3| = 3$$
Multiply:
$$2 \cdot 3 = 6$$
Inside the outer absolute value:
$$-7 + 6 = -1$$
$$|-1| = 1$$
The leading negative and multiply by 4:
$$-1 \cdot 4 = -4$$
Add 5:
$$-4 + 5 = 1$$
Answer: 1
Need help mastering Algebra 2 foundations?