7th Grade · 7.EE.B.4

Two-step equations in 7th grade

Two-step equations are equations that take two moves to solve, such as $4x + 7 = 31$ . In 7th grade, your child writes these equations from word problems and solves them. Your child also meets two-step inequalities, which have a whole range of answers instead of a single one.

What the standard says

The official Common Core wording, quoted exactly. The plain version follows.

7.EE.B.4 Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
7.EE.B.4.a Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach.
7.EE.B.4.b Solve word problems leading to inequalities of the form px + q > r or px + q r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem.

What it means, in plain words

A variable is a letter that stands for an unknown number. In a two-step equation, two things happen to the variable, usually a multiplication and then an addition or subtraction. Solving two-step equations means undoing those two things in reverse order: undo the addition first, then the multiplication. In $4x + 7 = 31$ , subtract 7 to get $4x = 24$ , then divide by 4 to get $x = 6$ . The numbers can also be negative, fractions or decimals.

Some equations have parentheses. For $3(x + 4) = 27$ , your child can divide both sides by 3 to get $x + 4 = 9$ , then subtract 4 to get $x = 5$ . Two-step equations in 7th grade are also compared with plain arithmetic. Without algebra, your child would reason backward: 27 shared by 3 is 9, and 9 minus 4 is 5. The steps are the same, in the same order.

Two-step inequalities use $<$ , $>$ , $\le$ or $\ge$ instead of an equals sign. They fit stories with words like "at most" or "at least". Your child solves them like equations, with one extra rule: multiplying or dividing both sides by a negative number flips the sign. The answers are then graphed on a number line, with an open circle when the end number is not included and a closed circle when it is. The answer also has to make sense in the story. If $x$ is a number of people, only whole numbers count.

Your child should be able to:

Write an equation or an inequality for a word problem and say what the variable stands for
Solve equations such as $px + q = r$ and $p(x + q) = r$ , including with fractions and decimals
Solve a two-step inequality and graph its answers on a number line
Explain what the answer means in the story

Example

A plumber charges a 40-dollar visit fee plus 55 dollars per hour. The bill was 205 dollars. How many hours did the plumber work?

Let $h$ be the number of hours. The equation is $55h + 40 = 205$ .
Subtract 40 from both sides: $55h = 165$ .
Divide both sides by 55: $h = 3$ .
Check: $55 \times 3 + 40 = 165 + 40 = 205$ . The plumber worked 3 hours.

Example

A soccer team has 90 dollars for new balls. Each ball costs 12 dollars, and shipping is 18 dollars. How many balls can the team buy?

Let $b$ be the number of balls. The total cost must be 90 dollars or less: $12b + 18 \le 90$ .
Subtract 18 from both sides: $12b \le 72$ .
Divide both sides by 12: $b \le 6$ .
On a number line, that is a closed circle at 6 with the shading going left. Only whole numbers make sense, so the team can buy any number of balls from 0 to 6.

Where children usually get stuck

For $5x + 8 = 43$ , your child divides 43 by 5 first and writes $x + 8 = 8.6$ .
Only the 43 got divided, and the 8 was left as it was. The 8 was added after the multiplication, so it has to come off first: $5x = 35$ , so $x = 7$ .
What helps. Ask what happened to $x$ , in order: times 5, then plus 8. Then undo it backward, the way you take off shoes before socks.
Your child writes $12b + 18 > 90$ for "the team can spend at most 90 dollars".
"Most" sounds like "more", so your child picks the greater-than sign. At most 90 means 90 or less.
What helps. Test a number in the story. Can the team spend 100 dollars? If the answer is no, the sign has to point the other way.
For $-2x + 5 > 11$ , your child gets $x > -3$ .
Dividing by a negative number reverses the order of numbers: 2 is less than 4, but $-2$ is greater than $-4$ . The sign has to flip, so the answer is $x < -3$ .
What helps. Check one number from the answer. Put $x = 0$ into $-2x + 5 > 11$ and you get $5 > 11$ , which is false. So $x > -3$ cannot be right.

Try four practice questions

Real 7th grade questions from our question bank, easiest first. Answer them right here, check each one, and see a worked solution for any question.

1Warm-up
Match each situation to the equation or inequality that models it. One model will not be used.
Connect each item on the left to its match on the right.
Situations
Models
T-shirts cost $7 each plus a $4 shipping fee. n is the number of shirts, and the total is $46.
Game cards cost $5 each. n is the number of packs, and after a $3 coupon the total is $32.
Juice boxes cost $3 each plus a $5 cooler fee. n is the number of boxes, and the total is at most $50.
$\displaystyle 4n + 7 = 46$
$\displaystyle 5n - 3 = 32$
$\displaystyle 7n + 4 = 46$
$\displaystyle 3n + 5 \leq 50$
Use Tab to navigate between items. Click or tap a card to select it, then click or tap a card on the opposite side to create a match. You can also drag a card or its connector dot to a card on the opposite side.
Show the answer and how to solve it
The matches are: T-shirts → 7n + 4 = 46, game cards → 5n − 3 = 32, and juice boxes → 3n + 5 ≤ 50. The unused model is 4n + 7 = 46.
1. 1In each situation, n is the number of items. The amount multiplied by n is the cost per item; the standalone number is a one-time change. Words like 'at most' mean ≤.
2. 2For the T-shirt situation, $7 repeats for each shirt (7n), and the $4 shipping fee is added once. The total is exactly $46.
3. 3For the game cards, $5 repeats per pack (5n). The $3 coupon is subtracted once. The total is exactly $32.
4. 4For the juice boxes, $3 repeats per box (3n), and the $5 cooler fee is added once. 'At most $50' means no more than $50, so use ≤.
5. 5The unused model is 4n + 7 = 46. It does not fit because it treats $4 as the repeated amount and $7 as the one-time amount, the reverse of the T-shirt situation.
Common mistake. A common mistake is swapping the coefficient and constant, writing 4n + 7 instead of 7n + 4 for the shirt situation. Another mistake is using = for 'at most $50' instead of ≤.
The idea underneath. The number multiplying the variable is the amount that repeats, the standalone number is a one-time change, and words like exactly, at least, and no more than choose the relation symbol.
2On level
Theo can spend at most $75 on birthday gifts. Look at the table and choose who is correct.
Student Choice Reason
Emma Inequality 75 is the highest amount he can spend.
Carlos Inequality Spending less than 75 also works.
A
Emma
B
Carlos
C
Both
D
Neither
Show the answer and how to solve it
The answer is Both because 'at most $75' sets a maximum, allowing a range of amounts, and both students correctly chose inequality with valid reasons.
1. 1Look at the table with Emma's and Carlos's ideas.
2. 2The phrase 'at most $75' sets a maximum. Theo can spend 75 or any smaller amount, which describes a range.
3. 3Emma chose inequality and said 75 is the highest amount he can spend, naming the upper limit. That correctly matches the situation, so Emma is right.
4. 4Carlos also chose inequality and said spending less than 75 also works, capturing the range of possible amounts. That is another valid way to describe the same situation, so Carlos is right too.
5. 5Both rows chose the correct model and gave valid reasons from different angles. The answer is Both.
Common mistake. A common mistake is picking only one student because the two reasons are worded differently. Both reasons correctly describe the same 'at most' inequality, one from the ceiling angle and one from the range angle.
The idea underneath. Use an equation for an exact amount in these situations. Use an inequality for a limit that allows a range of amounts.
3On level
A gym charges a $20 enrollment fee and $15 each month. The equation below shows that the total paid is $95. Which statement about x is correct?
$\displaystyle 15x + 20 = 95$
A
x represents the total paid, in dollars.
B
x represents the enrollment fee, in dollars.
C
x represents the monthly fee, in dollars.
D
x represents the number of months.
Show the answer and how to solve it
The correct statement is that x represents the number of months, because 15 is the monthly charge and 20 is the one-time enrollment fee.
1. 1Start with the modeled equation.
2. 2In 15x + 20 = 95, the +20 matches the one-time enrollment fee, and 95 matches the total paid. So x is not the enrollment fee or the total.
3. 3The term 15x means $15 is counted x times, so x tells how many monthly charges there are.
4. 4That means x represents the number of months. The units of x are months.
Common mistake. A common mistake is choosing the monthly fee because 15 is written next to x. But 15 is the amount charged each month; x tells how many months are being charged.
The idea underneath. In a context equation like px + q = r, the coefficient p is the amount for 1 unit of the variable, so the variable names the count or measurement being repeated.
4Stretch
Yara has $48 for the craft fair. Admission costs $6 and each supply kit costs $9. She plans to buy several kits. Which choice best tells whether to use an equation or an inequality, and why?
A
Equation, because the total cost must equal the $48 Yara brought.
B
Inequality, because the number of kits Yara buys is unknown.
C
Inequality, because the total cost can be $48 or less.
D
Equation, because the problem includes a fixed admission cost.
Show the answer and how to solve it
The answer is "Inequality, because the total cost can be $48 or less" because Yara's $48 is a spending cap — any total at or under $48 works.
1. 1The unknown is the number of supply kits Yara buys. The $6 admission is a one-time cost, and $9 is the cost for each kit.
2. 2Yara has $48 to spend. That is the most she can use — her total cannot go beyond $48, but she does not have to spend exactly $48.
3. 3Because several numbers of kits keep her total within $48 — for example, buying 1, 2, 3, or 4 kits all work — there is a range of acceptable totals, not just one.
4. 4When a situation allows a range of values instead of requiring one exact total, it is modeled with an inequality.
5. 5So the correct choice is: "Inequality, because the total cost can be $48 or less."
Common mistake. A common mistake is picking the equation choice because $48 looks like a total to match. Here, $48 is how much Yara has available, so her total can be any amount up to — but not exceeding — $48.
The idea underneath. To choose between an equation and an inequality, ignore how many numbers or operations appear and ask whether the situation requires one exact value or allows a range of values.

Student	Choice	Reason
Emma	Inequality	75 is the highest amount he can spend.
Carlos	Inequality	Spending less than 75 also works.