Grade 8 Workbook — Chapter 6: Multistep Linear Inequalities

SOL 8.PFA.5 · Companion to Textbook Chapter 6

Each page below is one Canva page. Headings are sized for direct paste: page title as H1, section labels as H2. Figures referenced by filename live in ../figures/. Every numbered item is the same problem as in the textbook, so one answer key serves both. Item numbers run continuously from 1 to 134.

PAGE 1 — Chapter opener

Chapter 6 · Multistep Linear Inequalities

Standard 8.PFA.5

In this chapter you will:

Solve linear inequalities that take up to four steps, with the variable on one or both sides
Expand with the distributive property and combine like terms before solving
Decide, at every step, whether the inequality symbol must reverse
Show every solution set two ways: algebraically and on a number line
Write inequalities from situations, and invent situations from inequalities
Test whether a particular number belongs to a solution set
Say what an answer means in the situation it came from

Words to know: inequality · strict · inclusive · solution set · multistep linear inequality · properties of inequality · distributive property · like terms · coefficient · constant term · reciprocal · reversal rule · boundary value · open circle · closed circle · ray · interpret in context

Reminder: reverse the symbol only when you multiply or divide both sides by a negative number.

PAGE 2 — What carries over

6.1 From Two Steps to Four

Fill in the blanks.

The symbols $<$ and $>$ are called ____________ inequalities. The boundary number is ____________ a solution.

The symbols $\le$ and $\ge$ are called ____________ inequalities. The boundary number ____ a solution.

To check an answer, substitute into the ____________ inequality, once with a value from ____________ the solution set and once with a value from ____________ it.

Guided practice.

Is $x = 6$ a solution of $3x - 4 \le 2x + 3$ ?

Left side: $3(\underline{\hspace{1cm}}) - 4 =$ ______ Right side: $2(\underline{\hspace{1cm}}) + 3 =$ ______

True or false? ____________ Solution? ______
Which values from $-1$ , $0$ , $2$ , and $5$ are solutions of $4x - 3 < 2x + 5$ ?

Solved form: ____________ Solutions: _______________

PAGE 3 — Guided practice, Lesson 6.1

Naming Every Step

Solve $3(x + 2) - 4 \le 11$ . Name the property beside each line.

Line	Property used

Solution: _______________

Solve $5x - 2x + 7 > 19$ .

Solution: _______________

Test inside, $x =$ ______ : _______________ true / false

Test outside, $x =$ ______ : _______________ true / false

PAGE 4 — Independent practice, Lesson 6.1

Practice · Solving and Testing

Solve.

a) $2x + 5x - 6 \le 15$	b) $4(x - 1) > 12$

c) $\dfrac{3}{4}x + \dfrac{1}{4}x - 2 \ge 6$	d) $9 - 3x + 5 \ge -1$

Which values from $-3$ , $0$ , $\tfrac{1}{2}$ , $4$ , and $7$ are solutions of $2(x - 1) + 3 \ge x + 4$ ?

Solved form: ____________ Solutions: _______________
List, in order, the operations you would use to solve $6x - 4 \le 2x + 8$ .

Step 1: ____________ Step 2: ____________ Step 3: ____________

Solution: ____________ Number of steps: ______
$\dfrac{1}{2}(4x + 6) < 11$ Solution: _______________
$0.5x + 1.5 \ge 2x - 3$ Solution: _______________

PAGE 5 — Application and error hunt, Lesson 6.1

Apply It · Find the Error

Name three solutions of $2x + 9 > 3$ , including one negative and one that is not an integer.

______ ______ ______ Show one check: _______________________________________________
Apply it. A rideshare charges a $3.50 pickup fee plus $1.25 per mile. Jordan will spend at most $21.

Let $m =$ _______________________________________________

Inequality: _______________ Solution: _______________

Greatest whole number of miles: ______
Find the error. Rosa solves $3x + 2x - 5 \le 20$ , gets $5x - 5 \le 20$ , then writes $5x \le 15$ and $x \le 3$ .

What went wrong? _______________________________________________

Correct solution: _______________

Test $x = 4$ in the original: _______________ What does it show? ____________

PAGE 6 — Exit ticket 6.1

Exit Ticket · Lesson 6.1

Name: ________________________ Date: ____________

$4x - x + 6 \le 18$ Solution: _______________
Is $x = -2$ a solution of $5x + 1 > 3x - 3$ ?

Left: ______ Right: ______ True or false? ______ Solution? ______
$2(x + 4) \ge 3x + 1$ Solution: _______________
What changes when an inequality takes more than two steps, and what stays exactly the same?

PAGE 7 — Why the symbol reverses

6.2 The Reversal Rule

FIGURE: fig1-reversal-reflection.png (full width)

FIGURE: fig2-slide-versus-reflect.png (full width)

Complete each sentence.

Multiplying both sides by a negative number ____________ every point across zero, so the order ____________.

Adding a negative number ____________ both points the same distance, so the order ____________.

Guided practice.

Start with $4 < 9$ . Multiply both sides by $-2$ .

Left: ______ Right: ______ True statement: _______________
Start with $4 < 9$ . Add $-12$ to both sides.

Left: ______ Right: ______ True statement: _______________ Direction changed? ______
Solve $-4x + 9 > 25$ . Solution: _______________

The direction reverses at this step: _______________________________________________
Solve $-\dfrac{2}{3}x - 5 \le 1$ . Solution: _______________ Boundary check: _______________

PAGE 8 — Independent practice, Lesson 6.2

Practice · Reverse or Not?

Solve.

a) $-5x + 3 \ge 18$	b) $7 - 2x < 1$

c) $-\dfrac{x}{4} + 2 \ge 5$	d) $-0.5x - 1.5 > 2$

Solve both. Only one reverses.

a) $x - 6 \ge -2$ : _______________ b) $-6x \ge -2$ : _______________

Why only one? _______________________________________________
$8 - 3(x + 1) > 14$ Solution: _______________
$12 - 5x \le 2x - 9$ Solution: _______________
Does the direction reverse? Circle one and give the reason.

Move	Reverses?	Reason
a) subtract $3x$ from both sides	yes / no
b) divide both sides by $-\tfrac{1}{2}$	yes / no
c) add $-7$ to both sides	yes / no
d) multiply both sides by $\tfrac{2}{5}$	yes / no

PAGE 9 — Application and error hunt, Lesson 6.2

Apply It · Find the Error

Apply it. A hot-air balloon at 1{,}200 feet descends 75 feet per minute. Its altitude after $t$ minutes is $1{,}200 - 75t$ feet.

Inequality for "altitude below 600 feet": _______________

Solution: _______________

Sentence answer: _______________________________________________
Explain. Start with $-6 < 2$ and multiply both sides by $-\tfrac{1}{2}$ .

True statement: _______________

Why did the direction have to reverse? _______________________________________________
Find the error. Nico solves $-2x + 5 \ge 11$ , gets $-2x \ge 6$ , and writes $x \ge -3$ .

Test $x = 0$ in the original: _______________ True or false? ______

What went wrong? _______________________________________________

Correct solution: _______________

PAGE 10 — Exit ticket 6.2

Exit Ticket · Lesson 6.2

Name: ________________________ Date: ____________

$-3x + 7 < -8$ Solution: _______________
$10 - \dfrac{x}{2} \ge 12$ Solution: _______________
Start with $-5 < 3$ and divide both sides by $-1$ . True statement: _______________
Why does distributing a negative number not reverse the symbol, while dividing both sides by a negative number does?

PAGE 11 — Expanding and combining

6.3 Clean Up Each Side First

FIGURE: fig4-distributive-solution.png (full width)

Distribute carefully. The outside factor multiplies every term inside, keeping its sign.

$-3(x + 2) = \underline{\hspace{2cm}}$ $-2(3x - 1) = \underline{\hspace{2cm}}$ $-(x - 5) = \underline{\hspace{2cm}}$

Guided practice.

$2(x - 3) + 5 < 13$

Line	Property used

Solution: _______________

$4(2x + 1) - 3x \ge 24$ Solution: _______________ Boundary check: _______________
$-3(x - 4) > 18$ Solution: _______________ Reversal happens at: _______________
$\dfrac{1}{2}(6x - 8) + 3 \le 11$ Solution: _______________

Inside test, $x =$ ______ : ______ Outside test, $x =$ ______ : ______

PAGE 12 — Independent practice, Lesson 6.3

Practice · Expand, Combine, Solve

Solve.

a) $3(x + 5) \le 27$	b) $-2(x - 7) < 4$

c) $5x - 2(x + 1) \ge 13$	d) $\dfrac{2}{3}(9x - 6) < 20$

$7 - 4(2x - 3) \le 27$ Solution: _______________
$6x + 3 - 2x - 11 > 8$ Solution: _______________
$0.25(8x - 4) \le 9$ Solution: _______________
$-\dfrac{1}{3}(6x + 9) > 5$ Solution: _______________

PAGE 13 — Application and error hunt, Lesson 6.3

Apply It · Find the Error

Apply it. Each gift bag holds a $4 pen and a $6 notebook. One flat $25 shipping charge covers the whole order. The club can spend at most $325.

Let $g =$ _______________________________________________

Inequality before combining: _______________

After combining like terms: _______________

Solution: _______________ Whole bags: ______
Explain. Solve $-2(x - 5) > 8$ : _______________

Two places a sign could go wrong here: _______________________________________________
Find the error. Ivy solves $-3(x + 2) \le 9$ by writing $-3x + 2 \le 9$ .

What went wrong? _______________________________________________

Correct solution: _______________

Test $x = -4$ in the original: ______ In Ivy's answer? ______

PAGE 14 — Exit ticket 6.3

Exit Ticket · Lesson 6.3

Name: ________________________ Date: ____________

$5(x - 2) + 4 \ge 19$ Solution: _______________
$9 - 2(x + 4) > 3$ Solution: _______________
$8x - 3(x - 1) \le 23$ Solution: _______________
Why can expanding and combining like terms never reverse the symbol, even when the factor is negative?

PAGE 15 — Variables on both sides

6.4 The Variable on Both Sides

FIGURE: fig5-both-sides-solution.png (full width)

The procedure.

Expand and ____________ like terms on each side.
Choose a side for the variable and ____________ that variable term on both sides.
Move the ____________ to the other side.
Divide by the ____________, and ask the reversal question.

Guided practice.

$5x - 3 > 2x + 9$

Line	Property used

Solution: _______________

$4x + 7 \le 9x - 8$ , solved two ways.

Subtract $4x$ first	Subtract $9x$ first

Same answer? ______   Which route reversed? ____________

PAGE 16 — Guided and independent practice, Lesson 6.4

Practice · Both Sides

$3(x - 2) < 5x + 4$ Solution: _______________ Boundary check: _______________
$6x - 4(x + 1) \le 3x + 5$ Solution: _______________

Inside test, $x =$ ______ : ______ Outside test, $x =$ ______ : ______
Solve.

a) $7x + 2 \ge 4x + 17$	b) $2x - 9 < 6x + 7$

c) $10 - x \le 3x - 6$	d) $\dfrac{1}{2}x + 4 > \dfrac{3}{2}x - 1$

$8 - 3x \ge 2(x + 9)$ Solution: _______________
$4(x + 1) < 2(3x - 5)$ Solution: _______________
$1.5x - 2 \le 0.5x + 3$ Solution: _______________
$9x - 5(x - 2) \ge 2x + 16$ Solution: _______________

PAGE 17 — Application and error hunt, Lesson 6.4

Apply It · Find the Error

Apply it. Gym A charges a $30 membership fee plus $5 per class. Gym B charges $10 per class with no fee.

Let $c =$ _______________________________________________

Inequality for "A costs less than B": _______________

Solution: _______________ Whole classes: _______________
Explain. Solve $3x + 8 \le 7x - 4$ two ways.

Subtract $3x$ first	Subtract $7x$ first

Which route reversed, and why do the answers still agree? _______________________________________________

Find the error. Marco solves $2x + 5 > 6x - 3$ , reaches $-4x > -8$ , and writes $x > 2$ .

Test $x = 3$ in the original: Left ______ Right ______ True or false? ______

Correct solution: _______________

PAGE 18 — Exit ticket 6.4

Exit Ticket · Lesson 6.4

Name: ________________________ Date: ____________

$6x - 1 \le 4x + 7$ Solution: _______________
$5 - 2x > 4x + 23$ Solution: _______________
$3(x + 4) \ge 5x - 2$ Solution: _______________
Why can moving the variable to the side that keeps its coefficient positive make a problem safer? Why is the other route still correct?

PAGE 19 — Graphing solution sets

6.5 Two Ways to Show One Answer

FIGURE: fig7-strict-vs-inclusive.png (full width)

FIGURE: fig6-four-solution-shapes.png (full width)

Complete the rules.

Open circle for the symbols ______ and ______. Closed circle for ______ and ______.

Shade ____________ for $>$ or $\ge$ . Shade ____________ for $<$ or $\le$ .

Graph the ____________ inequality, not the original one.

Guided practice. Use the blank number lines on the next page.

Graph $x > -3$ . Endpoint: ______ Circle: open / closed Shading: left / right
Graph $x \le \tfrac{5}{2}$ . Endpoint: ______ Circle: open / closed Shading: left / right

PAGE 20 — Graphing frames

Graph These

FIGURE: fig10-blank-number-lines.png (full width)

Use line a) for item 65, line b) for item 66, line c) for item 67.

Solve and graph $4x - 7 \ge 2x + 1$ .

Solution: _______________ Two values in the solution set: ______ ______
Solve and graph $-2x + 1 \ge 7$ .

Solution: _______________

Why does the graph run left? _______________________________________________

FIGURE: fig8-testing-values.png (full width)

PAGE 21 — Independent practice, Lesson 6.5

Practice · Solve and Graph

Graph each on your own number line. State endpoint, circle type, and direction.

	Endpoint	Circle	Direction
a) $x < 1$
b) $x \ge -5$
c) $x > \tfrac{1}{2}$
d) $x \le -2.5$

Solve and graph.

a) $3x + 5 \le x + 11$ : _______________

b) $9 - 2(x + 1) > 3x + 2$ : _______________
Solve and graph $-5(x - 1) \ge 20$ : _______________
Rewrite $12 \le 3x$ with the variable first: _______________ Then graph it.
A number line shows an open circle at $-1$ with shading to the left.

Inequality: _______________

A multistep inequality with the same solution set: _______________

Show that it does: _______________________________________________

PAGE 22 — Application and error hunt, Lesson 6.5

Apply It · Find the Error

Apply it. A pool holds 500 gallons and drains 20 gallons per minute, so it holds $500 - 20t$ gallons after $t$ minutes.

Inequality for "more than 100 gallons": _______________

Solution: _______________ Graph it.

Which part of the graph describes no real moment? _______________________________________________
Explain. Why does the graph of $-3x > 12$ run to the left?
Find the error. Priya solves $4 - x \le 9$ and graphs an open circle at $-5$ shaded left.

Error 1: ____________________ Error 2: ____________________

Test $x = 0$ in the original: ______ In Priya's graph? ______

Correct graph: _______________________________________________

PAGE 23 — Exit ticket 6.5

Exit Ticket · Lesson 6.5

Name: ________________________ Date: ____________

Solve and graph $7x - 4 < 5x + 6$ . Solution: _______________
Solve and graph $-\dfrac{1}{2}x + 3 \ge 5$ . Solution: _______________
A closed circle at $\tfrac{3}{2}$ shaded right. Inequality: _______________
State the open-circle and closed-circle rule, and say why the rule is what it is.

PAGE 24 — Inequalities in context

6.6 Write · Solve · Interpret · Invent

FIGURE: fig9-context-whole-weeks.png (full width)

Phrase table. Fill in the symbol.

at least, no fewer than	______	more than, exceeds	______
at most, no more than	______	fewer than, under	______

Guided practice.

A moving company charges $90 plus $40 per hour. A second charges $150 plus $25 per hour.

Let $h =$ _______________________________________________

Inequality for "the first is less expensive": _______________

Solution: _______________ Sentence answer: _______________________________________________
Five less than three times a number is at least twice the number increased by 4.

Inequality: _______________ Solution: _______________

PAGE 25 — Guided practice, Lesson 6.6

Interpreting the Answer

Devon has $120, buys a $12 book, then spends $15 each week. He wants more than $30 left.

Inequality: _______________ Solution: _______________

Test $w = 5$ : ______ Test $w = 6$ : ______ Whole weeks: ______
Write a situation in context for $8x + 20 \le 4x + 60$ .

Situation: _______________________________________________

Solution: _______________ What it means: _______________________________________________

PAGE 26 — Independent practice, Lesson 6.6

Practice · Context

Apply it. A freight elevator carries at most 1{,}500 pounds. A driver weighs 190 pounds, two helpers weigh 150 pounds each, and boxes weigh 45 pounds apiece.

Inequality: _______________ Solution: _______________ Whole boxes: ______
Apply it. Plan A: $25 per month plus $8 per gigabyte. Plan B: $55 per month plus $4 per gigabyte.

Inequality for "A costs no more than B": _______________ Solution: _______________

What it means: _______________________________________________
Four more than twice a number is less than the number decreased by 6.

Inequality: _______________ Solution: _______________
Write a situation for $12x + 45 \ge 165$ : _______________________________________________

Solution: _______________ What it means: _______________
Write a decreasing situation for $-6x + 80 > 20$ : _______________________________________________

Solution: _______________

PAGE 27 — Interpretation and error hunt, Lesson 6.6

Say What It Means · Find the Error

Interpretation. A crate problem gives $c \le 22.4$ , where $c$ counts whole crates.

In a sentence: _______________________________________________

Greatest usable value: ______

Check at that value: ______ Check at the next one: ______
Explain. In item 81, what does $h = 4$ mean, and why does the strict symbol exclude it?
Find the error. For "a printer costs $80 plus $0.20 per page, budget at most $140," Sam writes $80(0.2p) \le 140$ .

Why is that wrong? _______________________________________________

Correct inequality: _______________ Solution: _______________ Greatest number of pages: ______

PAGE 28 — Exit ticket 6.6

Exit Ticket · Lesson 6.6

Name: ________________________ Date: ____________

A caterer charges $200 plus $18 per guest; budget at most $920.

Inequality: _______________ Solution: _______________
Seven more than four times a number is at most the number increased by 25.

Inequality: _______________ Solution: _______________
Write a situation for $5x + 30 \le 8x$ : _______________________________________________

Solution: _______________ What it means: _______________
What does it mean to interpret a solution in context? Give an example where the algebra answer is not the practical answer.

PAGE 29 — Chapter 6 review, part 1

Chapter 6 Review

Part A · Solving multistep inequalities

Solve.

a) $3x + 4x - 9 \le 19$	b) $5(x - 3) > 10$

c) $-4x + 11 \ge 3$	d) $\dfrac{3}{5}x - 4 < 2$

$2(3x - 1) + 4x \le 38$ Solution: _______________
$8x - 3 > 5x + 12$ Solution: _______________
$7 - 2(x - 4) \le 3x$ Solution: _______________
$\dfrac{1}{4}(8x + 12) \ge x + 8$ Solution: _______________
$-0.5(4x - 6) < 9$ Solution: _______________
$6x - 5(x + 2) \ge 2x - 14$ Solution: _______________

PAGE 30 — Chapter 6 review, part 2

Chapter 6 Review (continued)

Part B · Algebraic and graphic representation

Graph each. State endpoint, circle type, and direction.

	Endpoint	Circle	Direction
a) $x \le 0$
b) $x > -4$
c) $x \ge \tfrac{7}{2}$
d) $x < 2.5$

Solve and graph $5x + 3 \le 3x - 7$ : _______________
Solve and graph $-3(x + 2) > 3$ : _______________
A closed circle at $-2$ shaded right.

Inequality: _______________ A multistep inequality with the same solution set: _______________

Show that it does: _______________________________________________
How does a graph show whether the boundary value is a solution? How would you confirm it algebraically?

PAGE 31 — Chapter 6 review, part 3

Chapter 6 Review (continued)

Part C · Writing an inequality from a situation

A bus holds at most 52 riders; 14 seats are taken. Inequality in $r$ : _______________
Three times a number decreased by 8 is greater than the number increased by 6.

Inequality: _______________ Solution: _______________
A gym charges $60 to join plus $12 per month; Priya can spend at most $240.

Inequality: _______________ Solution: _______________
Shop A: $35 setup plus $2.00 per shirt. Shop B: $75 setup plus $1.50 per shirt. For which numbers of shirts is A less expensive?

Inequality: _______________ Solution: _______________

Part D · Creating a situation from an inequality

Situation for $9x + 25 \le 160$ : _______________________________________________

Solution: _______________
Situation for $7x \ge 3x + 48$ : _______________________________________________

Solution: _______________

PAGE 32 — Chapter 6 review, part 4

Chapter 6 Review (continued)

Write a decreasing situation for $-8x + 100 > 20$ : _______________________________________________

Solution: _______________
Write a two-option comparison situation for $15 + 4x \le 6x - 5$ : _______________________________________________

Solution: _______________

Part E · Solving problems in context

A food truck charges $3.25 per taco plus a $1.50 packaging fee; Priya has $20.00.

Inequality: _______________ Solution: _______________ Tacos: ______
An elevator carries at most 2{,}000 pounds; the operator weighs 175 pounds and crates weigh 90 pounds each.

Inequality: _______________ Solution: _______________ Crates: ______
Mia has $85 and saves $22 per week. Noah has $250 and spends $8 per week. When does Mia have at least as much as Noah?

Inequality: _______________ Solution: _______________ Whole weeks: ______

PAGE 33 — Chapter 6 review, part 5

Chapter 6 Review (continued)

A 60-gallon tank drains 4 gallons per minute. When does it hold more than 12 gallons?

Inequality: _______________ Solution: _______________
Tickets sell for $8. The hall costs $240, and programs cost $3 per attendee. Profit must be at least $300.

Inequality: _______________ Solution: _______________ Tickets: ______

Part F · Identifying values in a solution set

Which of $-4$ , $0$ , $3$ , $5$ , $9$ solve $2(x - 1) \le x + 4$ ?

Solved form: ____________ Solutions: _______________
Which of $-6$ , $-3$ , $0$ , $2$ solve $-2x + 1 \ge 7$ ? Show each substitution.

Value	Substitution	True / false
$-6$
$-3$
$0$
$2$

Three solutions of $4x - 5 > x + 7$ , one not an integer: ______ ______ ______
A number that solves $3x + 2 \ge 11$ but not $3x + 2 > 11$ : ______ Why? ____________

PAGE 34 — Chapter 6 review, part 6

Chapter 6 Review (continued)

Part G · Interpreting solutions in context

$b \le 22.4$ , where $b$ counts whole boxes. Meaning: _______________________________________________

Greatest usable value: ______
$c > 6$ , where $c$ counts whole classes. Meaning: _______________________________________________

What happens at exactly 6? ____________
$t > 10$ seconds, altitude $240 - 15t$ meters. Meaning: _______________________________________________

Altitude at $t = 10$ : ______
$w \ge 8.5$ whole hours. Meaning: ____________ Why does this round up while 126 rounds down?

PAGE 35 — Chapter 6 review, part 7

Chapter 6 Review (continued)

Part H · Mixed application and reasoning

Solve and graph $-4(x - 2) \le 2x - 4$ . Solution: _______________ Boundary check: _______________
A student says $-2x + 3 > 9$ gives $x > -3$ . Test $x = 0$ : ______ True or false? ______

What does the test show? ____________ Correct solution: _______________
Write two multistep inequalities with solution set $x \le -1$ .

Variable on both sides: _______________ Variable on one side: _______________

Show each works: _______________________________________________
Solve $4x + 6 = 2x + 14$ : ______ Solve $4x + 6 \le 2x + 14$ : ______

How are the answers related? _______________________________________________

How do the graphs differ? _______________________________________________
Describe the whole procedure for deciding whether to reverse the symbol in a four-step inequality.

Example that reverses: _______________ Example with minus signs but no reversal: _______________

Canva production notes

Page size: 8.5 × 11 in, 0.75 in margins
Type: headings 24–28 pt, body 12–14 pt, answer blanks 14 pt with 1.5 line spacing
Number lines: fig10-blank-number-lines.png is the graphing frame; reprint it on any page where students are asked to graph, at least once per graphing item. Keep the tick spacing identical everywhere so students can compare graphs across pages.
Figure widths: every figure in this chapter is full width. fig6-four-solution-shapes.png is the tallest; give it its own block with no text wrapping beside it.
Color convention: blue accent = the solution set, red accent = a value being tested or rejected, matching the textbook figures. If pages print in grayscale, the open and closed circles still read correctly, so no extra instruction is needed.
Answer blanks: keep every blank on the same line as its prompt so the Canva text box does not reflow.