Signed Numbers and Absolute Value

Signed number errors are not conceptual. They are fast, careless, and they propagate: one sign error in the second line of a five-line problem produces an answer that is confidently wrong and often matches a distractor exactly.

Adding and subtracting

Think of a number line. A positive number moves right, a negative moves left.

Same signs: add the values and keep the sign.

• Negative 7 plus negative 5 is negative 12. Both moves go left.
• 7 plus 5 is 12.

Different signs: subtract the smaller absolute value from the larger, and take the sign of the larger.

• Negative 9 plus 4: the 9 is bigger, so subtract to get 5 and take the negative sign. The answer is negative 5.
• 9 plus negative 4 is positive 5.

Subtraction is addition of the opposite

This is the move that removes most sign errors. Rewrite every subtraction as an addition before you do anything else.

The third and fourth rows are where people lose points. Two minus signs in a row become a plus. Subtracting a negative is adding.

Multiplying and dividing

A completely different rule, and applying the addition rule here is the most common error in the topic.

Count the negative factors.

• An even count gives a positive result.
• An odd count gives a negative result.

Division follows the identical rule: negative 36 divided by negative 9 is positive 4.

Negatives and exponents

This is where the parentheses do real work.

• (-3) squared is 9. The parentheses mean the whole of negative 3 is being squared, so two negatives multiply to a positive.
• -3 squared is -9. With no parentheses, the exponent binds to the 3 only, and the minus sign is applied afterward. Read it as "the negative of 3 squared".

Both appear on the test and both are correct for what they say. When you write your own work, use parentheses so you can read it back.

The general pattern once the base itself is negative:

• A negative base to an even power is positive.
• A negative base to an odd power is negative.

So (-2) to the fourth is 16, and (-2) to the fifth is -32.

Absolute value

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The absolute value of a number is its distance from zero on the number line. Distance is never negative, which is the entire content of the idea.

• The absolute value of 7 is 7.
• The absolute value of -7 is 7.
• The absolute value of 0 is 0.

It is not "make it positive". It is "how far from zero", and that distinction matters as soon as anything else is attached.

The bars group, then the outside applies

Treat absolute value bars like parentheses: finish everything inside first, then take the distance, then deal with whatever is outside.

Row two is the one the test asks. A minus sign outside the bars survives, because the bars only govern what is inside them. An answer of 6 there is the standard error.

Row four is the other one: the bars are not distributive, so you cannot combine the insides first. The absolute value of (-9 plus -2) would be 11 as well here by coincidence of signs, but the absolute value of (-9 plus 2) is 7 while the sum of the absolute values is 11. Different questions.

What you can skip

Across the 55 questions on this topic:

• Absolute value equations. The absolute value questions evaluate the bars; none asks you to solve an equation with them.
• The word "absolute value" appears once; most questions just show the bars.

Where people lose points

Applying the multiplication rule to addition. Negative 5 plus negative 5 is negative 10, not positive 10.

Missing a double negative. Subtracting a negative adds. Rewrite subtraction as addition first.

Dropping a sign on distribution. Negative 2 times (x minus 4) is -2x plus 8, not -2x minus 8. The negative hits both terms.

Squaring a negative without parentheses and assuming the result is positive.

Treating absolute value as a sign eraser and killing a minus sign that was outside the bars.

Sign errors in the middle of a long problem. The defense is writing each line rather than doing two steps in your head.

Work one in under a minute

Evaluate: the negative of the absolute value of (-3 - 4), plus (-2) times (-5).

Inside the bars first: -3 minus 4 is -3 plus (-4), which is -7.

Absolute value of -7 is 7. The minus sign outside makes it -7.

Second term: (-2) times (-5) is two negatives, so positive 10.

Total: -7 plus 10 is 3.

Where this leads

Every algebra lesson that follows assumes signed arithmetic is automatic, and order of operations is where the absolute-value grouping rule is generalized.