Topic 1.2 · Unit 1 · CA CCSS A-SSE.1–2

Reading and building expressions

Two skills share this topic. One is mechanical, simplify correctly. The other is interpretive, say what a particular part of an expression means in the situation it describes. This course tests the second more than students expect, and it cannot be pattern-matched.

The method

Vocabulary you are expected to use exactly. In $ 5x^2 + 3x - 7 $: the terms are $ 5x^2 $, $ 3x $, $ -7 $; the coefficient of $ x^2 $ is 5; $ -7 $ is the constant. A factor is something multiplied, so $ 3(x+2) $ has factors 3 and $ (x+2) $.

To simplify: distribute across every bracket, remembering that a minus in front multiplies every term inside. Then combine like terms, identical variable parts only. $ 3x $ and $ 3x^2 $ are not like terms.
To interpret a part in context: ask what the whole expression counts, then what changes when that part changes. A number attached to a variable is a rate; a number standing alone is a starting value or one-off.
To test equivalence: simplify both and compare. One value agreeing is not proof; one value disagreeing is proof they differ.

Where marks are lost: the negative outside a bracket. $ 8 - 2(x - 3) $ is $ 8 - 2x + 6 = 14 - 2x $, not $ 8 - 2x - 6 $. The $ -2 $ multiplies the $ -3 $ as well, and two negatives make a plus.

Worked examples

Example 1: a negative outside a bracket. Simplify $ 4(2x + 5) - 3(x - 4) $.

$ 4(2x+5) = 8x + 20 $; $ -3(x-4) = -3x + 12 $. Note the $ +12 $, since $ -3 \times -4 = +12 $. Combining: $ 5x + 32 $.

Example 2: interpreting in context. A gym costs $ C = 45 + 28m $ after $ m $ months. What do 45 and 28 represent?

Ask what changes with $ m $. The 28 multiplies the months, so it is the cost per month, $28/month. The 45 is paid whatever $ m $ is, including $ m = 0 $, so it is the one-off joining fee. The giveaway is always whether the number is attached to the variable.

Example 3: reading a factored form. Weekly profit is $ P = 15(n - 40) $, where $ n $ is items sold. What does the 40 tell you?

Profit is zero when the bracket is zero, at $ n = 40 $, the break-even point. Below it the bracket is negative and the shop loses money. Multiplied out as $ 15n - 600 $ that fact is hidden, which is the whole reason the factored form is useful.

Practice · 15 problems

1–6 are direct practice, 7–12 mix the skills, and 13–15 ask you to interpret or diagnose rather than simplify.

Simplify $ -2(3x - 5) $.
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$ -2 \times 3x = -6x $; $ -2 \times -5 = +10 $. $ -6x + 10 $
Simplify $ 7x + 2y - 3x + 5y $.
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Only matching variable parts combine: $ 7x - 3x = 4x $, $ 2y + 5y = 7y $. $ 4x + 7y $
Simplify $ 8 - 2(x - 3) $.
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$ -2(x-3) = -2x + 6 $, then $ 8 + 6 = 14 $. $ 14 - 2x $. Writing $ 8 - 2x - 6 $ is the standard error.
Simplify $ 3x^2 + 2x - x^2 + 5x $.
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$ x^2 $ terms: $ 3 - 1 = 2 $. $ x $ terms: $ 2 + 5 = 7 $. They cannot combine with each other. $ 2x^2 + 7x $
Simplify $ \frac{1}{2}(6x - 8) + 3 $.
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$ 3x - 4 $, then $ -4 + 3 = -1 $. $ 3x - 1 $
Simplify $ 2(3x - 1) - (x + 5) $.
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A bare minus before a bracket is $ -1 \times $ everything inside: $ -(x+5) = -x - 5 $. With $ 6x - 2 $: $ 5x - 7 $
In $ 7x^2 - 4x + 9 $, name the coefficient of $ x $ and the constant term.
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The coefficient of $ x $ is $ -4 $, the sign belongs to it. The constant is 9. $ -4 $ and 9
Are $ 3(x + 4) $ and $ 3x + 4 $ equivalent? Justify.
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No. $ 3(x+4) = 3x + 12 $. One counterexample settles it: at $ x = 0 $ the first gives 12 and the second gives 4. Not equivalent
Write an expression for the perimeter of a rectangle of width $ w $ whose length is 3 more than twice the width.
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Length $ = 2w + 3 $. Perimeter $ = 2w + 2(2w+3) = 2w + 4w + 6 $. $ 6w + 6 $
Simplify $ 3(2x - 5) - 4(x - 2) + 7 $.
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$ 6x - 15 $, $ -4x + 8 $, $ +7 $. Then $ 6x - 4x = 2x $ and $ -15 + 8 + 7 = 0 $. $ 2x $
A taxi charges $4 plus $2.50 per mile. Write an expression for an $ m $-mile trip.
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The per-mile rate attaches to the variable; the flag-fall stands alone. $ 4 + 2.5m $
Show that $ 2(x + 3) + 4x $ and $ 6(x + 1) $ are equivalent.
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Left: $ 2x + 6 + 4x = 6x + 6 $. Right: $ 6x + 6 $. Identical after simplifying, so equivalent for every $ x $. Both give $ 6x + 6 $
A phone plan costs $ C = 20 + 0.05t $ dollars for $ t $ minutes. Interpret both numbers, with units.
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0.05 attaches to $ t $, so it is 5 cents per minute. 20 is paid even at $ t = 0 $, so it is the fixed monthly fee. $20 fixed; $0.05 per minute
Revenue is $ R = 12(s - 30) $ for $ s $ items sold. What is significant about $ s = 30 $, and why is this form more useful than $ 12s - 360 $?
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At $ s = 30 $ the bracket is zero so $ R = 0 $, break-even. The factored form shows it directly; the expanded form requires solving to find it. Break-even at 30 items
A student simplifies $ 5 - 3(2x - 4) $ to $ 5 - 6x - 12 = -6x - 7 $. Find the error.
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They distributed $ -3 $ to the $ 2x $ correctly but kept the sign on the $ -4 $. $ -3 \times -4 = +12 $, so it is $ 5 - 6x + 12 $. $ 17 - 6x $

Reading and building expressions

Keep going in the full Integrated Mathematics 1 course