Algebra 1 Workbook — Chapter 3: Linear Inequalities in One Variable

SOL A.EI.1 (c, f) · Companion to Textbook Chapter 3

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 88.

PAGE 1 — Chapter opener

Chapter 3 · Linear Inequalities in One Variable

Standard A.EI.1 (c, f)

In this chapter you will:

Solve multistep linear inequalities in one variable and name the property behind every step
Graph each solution set on a number line, with the right endpoint and the right direction
Decide, at every step, whether the inequality symbol must reverse — and explain why
Write and solve inequalities for real situations, including situations where something decreases
Verify a solution set three ways: algebraically, graphically, and with technology
Say what an answer means in the situation it came from, and whether the endpoint is usable

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

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

PAGE 2 — What an inequality asks

3.1 A Point, or a Region?

FIGURE: fig1-equation-versus-inequality.png (full width)

FIGURE: fig2-open-versus-closed.png (full width)

Fill in the blanks.

The symbols $<$ and $>$ are called ____________ inequalities, and the boundary number ____ a solution.

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

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

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

Guided practice.

Is $x = -2$ a solution of $4x + 7 \le 3x + 5$ ?

Left side: $4(\underline{\hspace{1cm}}) + 7 =$ ______ Right side: $3(\underline{\hspace{1cm}}) + 5 =$ ______

Number sentence: _______________ True or false? ______ Solution? ______

PAGE 3 — Guided practice, Lesson 3.1

Naming Every Step

Solve and graph $3x - 5 < 7$ .

Solution: _______________ Endpoint: ______ Circle: open / closed Shading: left / right
Solve $2x + 1 \ge 7$ . Name the property beside each line.

Line	Property used

Solution: _______________

Solve $5x - 3 > 2x + 9$ , naming the property at each step.

Line	Property used

Solution: _______________ Boundary test, $x = 4$ : ______ True / false: ______

PAGE 4 — Graphing frames, Lesson 3.1

Graph These

FIGURE: fig8-blank-number-lines.png (full width)

Use line a) for item 2 and line b) for item 3. Lines c) and d) are for items 5a and 5b; reprint the frame for 5c and 5d.

Solve and graph.

a) $x + 6 \le 10$	b) $7x > 21$

c) $4x - 9 \ge 11$	d) $\dfrac{1}{3}x + 2 < 5$

$6x + 5 \le 4x + 17$ Solution: _______________ Property used to move the $4x$ : _______________
Rewrite $15 < 5x$ with the variable first: _______________ Then graph it.

PAGE 5 — Independent practice, Lesson 3.1

Practice · Sets and Sentences

Which values from $-5$ , $0$ , $2.5$ , $4$ , and $9$ are solutions of $3x - 4 \le x + 6$ ?

Solved form: ____________ Solutions: _______________
A number line shows a closed circle at $-2$ with shading to the right.

Inequality: _______________

A multistep inequality with the same solution set: _______________

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

______ ______ ______ Show one check: _______________________________________________

PAGE 6 — Application and reasoning, Lesson 3.1

Apply It · Explain It

Apply it. A robotics club has $\$120$ . A base kit costs $\$45$ , and each sensor costs $\$5$ .

Let $s =$ _______________________________________________

Inequality: _______________ Solution: _______________

Greatest whole number of sensors: ______ Check at that value: _______________
Explain. Why is the answer to $2x + 1 = 7$ a single point, while the answer to $2x + 1 \ge 7$ is a ray?

Where does the number 3 come from in both problems? _______________________________________________

PAGE 7 — Exit ticket 3.1

Exit Ticket · Lesson 3.1

Name: ________________________ Date: ____________

$5x - 8 < 2$ Solution: _______________ Circle: open / closed Shading: left / right
Is $x = 3$ a solution of $4x - 5 \ge 2x + 1$ ?

Left: ______ Right: ______ True or false? ______ Solution? ______
$8x + 3 \ge 5x + 18$ Solution: _______________
State the open-circle and closed-circle rule, and say why the rule is what it is.

PAGE 8 — Why the symbol reverses

3.2 The Reversal Rule

FIGURE: fig3-reversal-reflection.png (full width)

FIGURE: fig4-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 $2 < 5$ . Multiply both sides by $-3$ .

Left: ______ Right: ______ True statement: _______________ Direction changed? ______
Start with $2 < 5$ . Add $-8$ to both sides.

Left: ______ Right: ______ True statement: _______________ Direction changed? ______

PAGE 9 — Guided practice, Lesson 3.2

Reverse, or Leave It Alone?

FIGURE: fig5-solve-and-graph-reversal.png (full width)

Solve and graph $-3x + 7 \ge 22$ .

Solution: _______________

The direction reverses at this step: _______________________________________________
Solve $-\dfrac{2}{5}x < 6$ . Solution: _______________

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

PAGE 10 — Independent practice, Lesson 3.2

Practice · Reverse or Not?

Solve and graph.

a) $-5x > 20$	b) $-x + 3 \le 10$

c) $-\dfrac{1}{2}x - 4 \ge 1$	d) $8 - 3x < 23$

Solve both. Only one reverses.

a) $x - 8 \ge -3$ : _______________ b) $-8x \ge -3$ : _______________

Why only one? _______________________________________________
$6 - 2(x + 4) > 10$ Solution: _______________
$4 - 7x \le 3x - 26$ Solution: _______________ Boundary check: _______________
Does the direction reverse? Circle one and give the reason.

Move	Reverses?	Reason
a) subtract $5x$ from both sides	yes / no
b) multiply both sides by $-\tfrac{3}{4}$	yes / no
c) distribute $-2$ across $(x - 3)$	yes / no
d) divide both sides by $0.5$	yes / no

PAGE 11 — Application and error hunt, Lesson 3.2

Apply It · Find the Error

Apply it. A drone hovers at 320 feet and descends 25 feet per second, so its altitude after $t$ seconds is $320 - 25t$ feet.

Inequality for "below 145 feet": _______________

Solution: _______________

Sentence answer: _______________________________________________
Explain. Start with $-8 < -2$ and multiply both sides by $-\tfrac{1}{2}$ .

True statement: _______________

Why did the direction have to reverse? _______________________________________________
Find the error. Kai solves $-5x + 2 > 17$ , reaches $-5x > 15$ , and writes $x > -3$ .

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

What went wrong? _______________________________________________

Correct solution: _______________

PAGE 12 — Exit ticket 3.2

Exit Ticket · Lesson 3.2

Name: ________________________ Date: ____________

$-4x + 5 \le -11$ Solution: _______________ Circle: open / closed Shading: left / right
$10 - \dfrac{x}{3} > 12$ Solution: _______________
Start with $-9 < 6$ and divide both sides by $-3$ . True statement: _______________
Why does multiplying both sides by a negative number reverse the symbol, while distributing a negative number does not?

PAGE 13 — Multistep work

3.3 Clean Up, Collect, Isolate

The order of work.

____________ every product written with parentheses.
____________ like terms on each side.
____________ the variable on one side.
Move the ____________ to the other side.
Divide by the ____________, and ask the reversal question.

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

$-3(x + 2) = \underline{\hspace{2cm}}$ $-2(x + 6) = \underline{\hspace{2cm}}$ $-4(x - 3) = \underline{\hspace{2cm}}$

Guided practice.

$3(2x - 5) + 4 \le 25$

Line	Property used

Solution: _______________

$5x - 2(x + 6) > 3$ Solution: _______________ Boundary check, $x = 5$ : _______________

PAGE 14 — Guided practice, Lesson 3.3

Two Routes, One Answer

Solve $7x + 4 \ge 10x - 11$ two ways.

Subtract $7x$ first	Subtract $10x$ first

Same answer? ______   Which route reversed? ____________   Why? ____________________

Solve and graph $2(3 - x) \ge 4(x + 3) - 6$ .

Solution: _______________ Endpoint: ______ Circle: open / closed Shading: left / right

PAGE 15 — Independent practice, Lesson 3.3

Practice · Expand, Combine, Solve

Solve and graph.

a) $4(x - 3) < 8$	b) $-2(x + 5) \ge 4$

c) $6x - 3(x - 2) \le 18$	d) $\dfrac{2}{3}(9x - 6) > 20$

$8 - 5(2x - 1) \ge -17$ Solution: _______________ Property at the reversal step: _______________
$9x - 4 < 5x + 12$ Solution: _______________
$0.4x + 1.2 \ge 0.9x - 0.8$ Solution: _______________
$3(x + 2) - 4x \le 2(4 - x)$ Solution: _______________

PAGE 16 — Error hunt, application, reasoning, Lesson 3.3

Find the Error · Apply It · Explain It

Find the error. Dana solves $-4(x - 3) \le 20$ by writing $-4x - 12 \le 20$ .

What went wrong? _______________________________________________

Correct solution: _______________ Dana's answer: _______________

Test $x = -3$ in the original: ______ Is $-3$ in Dana's set? ______
Apply it. Hall A charges $\$250$ plus $\$22$ per guest. Hall B charges $\$400$ plus $\$16$ per guest.

Let $g =$ _______________________________________________

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

What it means: _______________________________________________
Explain. Solve $7 - 3(x - 2) \ge 4x - 8$ : _______________

Two places a sign could go wrong here: _______________________________________________

PAGE 17 — Exit ticket 3.3

Exit Ticket · Lesson 3.3

Name: ________________________ Date: ____________

$6(x - 1) + 2 \le 20$ Solution: _______________
$12 - 4(x + 1) > 2x - 4$ Solution: _______________ Circle: open / closed Shading: left / right
$5x - 7 \ge 9x + 5$ Solution: _______________
Why can expanding and combining like terms never reverse the symbol, even when the factor is negative?

PAGE 18 — Three ways to be sure

3.4 Verify · Explain · Interpret

FIGURE: fig6-graphical-verification.png (half width, right)

The three verifications.

Algebraically: substitute into the ____________ inequality — one value ____________ the set, the ____________ itself, and one value ____________ it.

Graphically: graph each side as a line. They cross at the ____________. The solution set is where the correct line is ____________.

With technology: enter the two sides as $Y_1$ and $Y_2$ , use ____________ to find the boundary, then read the ____________ to decide the direction.

Guided practice.

Verify that $4x - 7 \ge 2x + 1$ solves to $x \ge 4$ .

Inside, $x =$ ______ : ______ Boundary, $x = 4$ : ______ Outside, $x =$ ______ : ______
Verify graphically that $2x - 1 > x + 1$ solves to $x > 2$ .

The lines cross at ( ______ , ______ ). Higher to the right of the crossing: _______________

Why is the crossing point excluded? _______________________________________________

PAGE 19 — Technology and context, Lesson 3.4

Calculator Check · Say What It Means

FIGURE: fig7-context-whole-tables.png (full width)

Verify $-3x + 7 \ge 22$ with a graphing calculator.

$Y_1 =$ ____________ $Y_2 =$ ____________ Intersection: ( ______ , ______ )

Which side of the crossing is $Y_1$ higher? ____________ Solution: _______________

What the table row at the boundary shows: _______________________________________________
Apply it. A student council has $\$500$ . The DJ costs $\$180$ and decorations cost $\$12$ per table.

Let $t =$ _______________________________________________

Inequality: _______________ Solution: _______________

Test $t = 26$ : ______ Test $t = 27$ : ______ Whole tables: ______

PAGE 20 — Independent practice, Lesson 3.4

Practice · Verify Three Ways

$3x + 2 \le 5x - 8$ Solution: _______________

Inside, $x =$ ______ : ______ Boundary: ______ Outside, $x =$ ______ : ______
$-2x + 9 > 1$ Solution: _______________

The lines $y = -2x + 9$ and $y = 1$ cross at ( ______ , ______ ). Higher to the left: ____________
Use the table to decide where $5 - 2x \ge x - 4$ .

$x$	0	1	2	3	4	5
$Y_1 = 5 - 2x$	5	3	1	$-1$	$-3$	$-5$
$Y_2 = x - 4$	$-4$	$-3$	$-2$	$-1$	0	1

From the table: _______________   Solved algebraically: _______________   Do they agree? ______

PAGE 21 — Application and interpretation, Lesson 3.4

Apply It · Say What It Means

Apply it. A phone plan costs $\$35$ per month plus $\$0.05$ per extra text. Amara will spend at most $\$50$ .

Inequality: _______________ Solution: _______________ What it means: ____________________
Apply it. An elevator carries at most 2{,}000 pounds. The operator weighs 180 pounds; crates weigh 65 pounds each.

Inequality: _______________ Solution: _______________

Why is the endpoint usable here? _______________________________________________
Interpretation. A subscription problem gives $m \le 13.75$ , where $m$ counts whole months.

In a sentence: _______________________________________________

Greatest usable value: ______ Check at that value: ______ Check at the next one: ______

PAGE 22 — Reasoning and error hunt, Lesson 3.4

Explain It · Find the Error

Explain. Why is one inside test not enough to verify a solution set?

The three tests that are convincing: ____________ , ____________ , ____________

Which one decides open versus closed? ____________
Find the error. Priya solves $6 - 2x \ge 14$ and reports $x \ge -4$ .

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

What went wrong? _______________________________________________

Correct solution: _______________ Correct graph: _______________________________________________

PAGE 23 — Exit ticket 3.4

Exit Ticket · Lesson 3.4

Name: ________________________ Date: ____________

$4 - 3x < 19$ Solution: _______________

Inside: ______ Boundary: ______ Outside: ______
To check item 61 graphically, graph $Y_1 =$ ____________ and $Y_2 =$ ____________ .

They cross at ( ______ , ______ ). Which side is $Y_1$ lower? ____________
Apply it. A food truck charges $\$2.75$ per taco plus a $\$3.00$ packaging fee. Devon has $\$25.00$ .

Inequality: _______________ Solution: _______________ Tacos: ______
What does it mean to interpret a solution in context? Give an example where the algebraic endpoint is not the practical answer.

PAGE 24 — Chapter 3 review, part 1

Chapter 3 Review

Part A · Solving and graphing (A.EI.1c)

Solve and graph. State the endpoint, circle type, and direction.

	Solution	Endpoint	Circle	Direction
a) $5x - 3 \ge 12$
b) $2x + 11 < 3$
c) $-6x \le 18$
d) $\dfrac{x}{4} - 1 > 2$

$4(x - 2) + 9 \le 17$ Solution: _______________

Line	Property used

$7 - 2(x + 3) > 5$ Solution: _______________

PAGE 25 — Chapter 3 review, part 2

Chapter 3 Review (continued)

$8x + 1 \ge 3x - 14$ Solution: _______________
$-\dfrac{3}{4}x + 2 < -1$ Solution: _______________
$5(2x - 3) \le 3(3x + 1)$ Solution: _______________
$6 - x \ge 4x - 9$ Solution: _______________
Rewrite $20 > 5x$ with the variable first: _______________ Then graph it.
Which of $-8$ , $-3$ , $0$ , $2.5$ , $7$ solve $-2x + 5 \ge 11$ ?

Solved form: ____________ Solutions: _______________
An open circle at $\tfrac{3}{2}$ shaded right.

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

Show that it does: _______________________________________________

PAGE 26 — Chapter 3 review, part 3

Chapter 3 Review (continued)

Apply it. A hiker starts at 2{,}400 feet and descends 150 feet per hour, so her elevation after $h$ hours is $2{,}400 - 150h$ feet.

Inequality for "below 1{,}500 feet": _______________ Solution: _______________

Graph it. Which part of the graph describes no real moment? _______________________________________________
Explain. State the complete rule for reversing the symbol.

An example that reverses: _______________ Solution: ____________

An example with several minus signs and no reversal: _______________ Solution: ____________

PAGE 27 — Chapter 3 review, part 4

Chapter 3 Review (continued)

Part B · Verifying and interpreting (A.EI.1f)

$9 - 4x \le 25$ Solution: _______________

Inside: ______ Boundary: ______ Outside: ______
Verify graphically that $3x - 2 < x + 4$ solves to $x < 3$ .

The two lines: ____________ and ____________ They cross at ( ______ , ______ )

Which line is lower to the left of the crossing? ____________
Use the table to estimate where $2x + 1 \ge 8 - x$ .

$x$	0	1	2	3	4	5
$Y_1 = 2x + 1$	1	3	5	7	9	11
$Y_2 = 8 - x$	8	7	6	5	4	3

From the table, the boundary is between ______ and ______ .   Solved algebraically: _______________

Why can the table not give the exact boundary? _______________________________________________

How would you verify a solution set with a graphing calculator, and what would you do if the calculator and your algebra disagreed?

PAGE 28 — Chapter 3 review, part 5

Chapter 3 Review (continued)

Apply it. Gym A: $\$40$ per month plus $\$6$ per class. Gym B: $\$70$ per month plus $\$2$ per class.

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

Whole classes: ______
Apply it. A rain barrel holds 90 gallons and loses 6 gallons per day, so it holds $90 - 6d$ gallons after $d$ days.

Inequality for "more than 30 gallons": _______________ Solution: _______________

Which part of the graph describes no real day? _______________________________________________
Apply it. A delivery service charges a $\$12.00$ handling fee plus $\$7.50$ per box. A shop will spend at most $\$60.00$ .

Inequality: _______________ Solution: _______________ Whole boxes: ______

PAGE 29 — Chapter 3 review, part 6

Chapter 3 Review (continued)

Interpretation. Item 57 gives $c \le 28$ crates; item 83 gives $b \le 6.4$ boxes.

Why is the endpoint part of the answer in one and not the other? _______________________________________________

The test that settles each: _______________________________________________
Interpretation. A descent gives $h > 6$ hours, with elevation $2{,}400 - 150h$ feet.

In a sentence: _______________________________________________

Elevation at exactly $h = 6$ : ______ Why is $h = 6$ excluded? ____________________
Explain. A classmate verifies $x > 5$ by testing only $x = 10$ .

Why is that incomplete? _______________________________________________

Tests he is missing: ____________________

A wrong answer his test would not catch: _______________

PAGE 30 — Chapter 3 review, part 7

Chapter 3 Review (continued)

Find the error. Marcus solves $5 - 4x > 21$ , reaches $-4x > 16$ , and writes $x > -4$ .

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

Correct solution: _______________ Correct graph: _______________________________________________
Write a situation in context for $25 + 4x \le 81$ .

Situation: _______________________________________________

Solution: _______________ Boundary test: ______ Outside test: ______

What it means: _______________________________________________

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: fig8-blank-number-lines.png is the graphing frame. It appears once, on page 4, but must be reprinted on every page that asks a student to graph — pages 9, 10, 14, 15, 17, 24, 25, and 26 all contain graphing items. Keep the tick spacing identical on every reprint so students can compare graphs across pages.
Figure widths: every figure is full width except fig6-graphical-verification.png, which is tall and narrow — set it at half width on the right of page 18 with the text wrapping at its left. fig3-reversal-reflection.png is the widest; give it its own block with no text beside it.
Work space: leave at least 1.5 in of blank space below every "Solve and graph" table cell. The reversal items on pages 10 and 11 need the most room, because students are asked to show the step where the symbol turns.
Color convention: blue accent = the solution set, red accent = a value being tested or rejected, matching the textbook figures. Open and closed circles still read correctly in grayscale, so no extra instruction is needed for black-and-white printing.
Answer blanks: keep every blank on the same line as its prompt so the Canva text box does not reflow.