SHSAT: Expressions: distribute, combine and model
Simplify expressions with parentheses and negative signs, evaluate them with negative numbers, and write expressions from word descriptions.
How do you simplify or write an expression without losing a sign or a term?
Short answer
Distribute every number, including its sign, to each term inside the parentheses, then combine like terms. Check your result by substituting a small number.
Know first: Multiplying negative numbers, Like terms
Step 1 of 5
A minus sign in front of parentheses reaches every term
To simplify an expression, distribute each number to every term inside its parentheses, then combine like terms: x terms with x terms, plain numbers with plain numbers.
The sign in front of the parentheses comes along. −4(x + 6) is −4x − 24, and −(2x − 7) is −2x + 7. A common slip is dropping that sign on the second term.
| Expression | Equals | Why |
|---|---|---|
| −4(x + 6) | −4x − 24 | −4 times +6 is −24 |
| −(2x − 7) | −2x + 7 | a minus times −7 is +7 |
| 2a² when a = −3 | 18 | square first: (−3)² = 9, then × 2 |
| −ab when a = −3, b = 5 | 15 | ab = −15, and −(−15) = 15 |
When you substitute a negative number, wrap it in parentheses.
Step 2 of 5
Distribute, then combine
Worked example: Two sets of parentheses
Which expression is equivalent to 5(x − 2) − 2(3x − 4)?
- A−x − 18
- B−x − 14
- C−x − 2
- D11x − 18
- 1
Distribute the 5
5x − 10.
- 2
Distribute the −2
−2 × 3x = −6x and −2 × (−4) = +8, so −6x + 8.
- 3
Combine like terms
5x − 6x = −x and −10 + 8 = −2. The result is −x − 2.
- 4
Check with a number
At x = 1: 5(−1) − 2(−1) = −5 + 2 = −3, and −1 − 2 = −3.
- 5
See why the other choices are there
−x − 18 makes −2 × (−4) negative. −x − 14 multiplies only the 3x by −2 and leaves −4. 11x − 18 adds 6x instead of subtracting and also gets the sign of 8 wrong.
Answer
−x − 2
Find the wrong step
Evaluate 3a² − ab when a = −2 and b = 4.
- 1
3a² = (3 × −2)² = 36.
What went wrong
The exponent belongs only to a. Square −2 first, then multiply by 3: 3 × 4 = 12.
- 2
ab = (−2)(4) = −8, so −ab = +8.
- 3
36 + 8 = 44.
The fix
3a² = 3 × (−2)² = 12, and 12 + 8 = 20.
Step 3 of 5
Simplify, evaluate and model
Check yourself · Question 1
Which expression is equivalent to 6(x − 1) − (2x − 5)?
Answer: B
6x − 6 − 2x + 5 = 4x − 1. The minus sign in front of (2x − 5) makes −5 into +5.
Trap answer: 4x − 11
Distributing the minus to only one term
- Why it looks right
- The minus sign sits right next to 2x, so it's easy to apply it there and stop.
- Why it's wrong
- A minus in front of parentheses multiplies every term inside by −1. −(2x − 5) = −2x + 5.
- The right answer
- 4x − 1, from 6x − 6 − 2x + 5.
Check yourself · Question 2
The expression 5x + c equals 23 when x = 4. What is its value when x = −3?
Answer: C
First find c: 5(4) + c = 23, so c = 3. Then 5(−3) + 3 = −15 + 3 = −12.
Check yourself · Question 3
Pens cost p dollars each. A pack of 5 pens gets a single $2 discount on the whole pack, and a notebook costs $4. Which expression gives the total cost of one pack and one notebook?
Answer: B
Five pens cost 5p. The $2 discount happens once: 5p − 2. Add the notebook: 5p − 2 + 4 = 5p + 2.
Common mistake
I let the minus sign touch only the first term in the parentheses.
- Why it's tempting
- When you read left to right, the minus seems to belong to the term right after it.
- Do this instead
- Rewrite −(2x − 5) as −1(2x − 5) and multiply both terms. Then check your expression by substituting a small number such as x = 1.
Step 4 of 5
What to remember
Remember
- Distribute to every term, carrying the sign in front of the parentheses.
- Square a negative number inside parentheses before multiplying by a coefficient.
- In a model, decide whether each amount happens once or once per item, then check with a small number.
Step 5 of 5
Practice on a real question
Use what you just learned on this question, then check the explanation.
In your own words
In the practice question, what did −3(x + 5) become after you distributed, and which choice would you get if the last term stayed positive?