Linear Equations and Inequalities

⚖️ Linear Equations and Inequalities

Part 1 of 7 — Algebra on the ASVAB: Expressions and Substitution

Topic Overview

What This Looks Like on the ASVAB

Mathematics Knowledge (MK) is the fifth subtest on the CAT-ASVAB, right after Paragraph Comprehension. You get 15 scored questions in 31 minutes, so about 2 minutes per question. Every question has four answer choices, and no calculator is allowed. At MEPS you get two sheets of scratch paper and two pencils, and you can ask for more.

The official description of MK includes "algebraic operations and equations," and one of the official sample topics is solving an equation like $x + a = b$. Expect several MK questions that come down to the skills in this lesson:

• finding the value of an expression when you are told what the letters equal
• solving an equation for $x$
• solving an inequality and picking the matching graph
• writing an equation from a sentence
• solving two equations together

MK is one of the four AFQT subtests (AR, MK, WK and PC). Your AFQT is built from AR + MK + 2·VE, so every MK point you pick up here helps decide whether you can enlist, not just which jobs you qualify for.

Good news: algebra on the ASVAB is the same handful of moves, over and over. Learn the moves and the questions start to look familiar.

Warm-up. 🎯 Three quick questions to see where you are. No pressure: this is just a starting point.

The Words Algebra Uses

A variable is a letter that stands for a number you don't know yet. In $7y - 4$, the variable is $y$.

Some shorthand to know cold:

• $3x$ means $3 \cdot x$. A number written next to a letter is multiplied.
• $x$ by itself means $1x$.
• $\frac{x}{4}$ means $x \div 4$.
• $2(x + 5)$ means $2$ times the whole group $x + 5$.

Combining like terms

Like terms have the same variable part. You add or subtract their coefficients and leave the variable alone:

$6x - 2 - 4x + 9 = (6x - 4x) + (-2 + 9) = 2x + 7$

$3x$ and $3$ are not like terms: one has a variable and one doesn't, so $3x + 3$ stays as it is.

The distributive property

A number in front of parentheses multiplies every term inside:

$3(x + 4) = 3x + 12 \qquad -2(x - 5) = -2x + 10$

Watch the second one: $-2$ times $-5$ is positive $10$. Dropping that sign is one of the most common mistakes on the whole MK subtest.

Substitution: Plugging In a Value

"Evaluate" means: replace each letter with its number, then do the arithmetic in order (parentheses, exponents, multiply and divide, add and subtract).

Always put the number in parentheses when you substitute. It protects negative signs.

Worked example 1

Find $5 - 3m$ when $m = -2$.

$5 - 3(-2) = 5 - (-6) = 5 + 6 = 11$

Subtracting a negative is adding.

Worked example 2

Find $2a^2 - 3b$ when $a = -3$ and $b = 4$.

1. Substitute with parentheses: $2(-3)^2 - 3(4)$
2. Exponent first: $(-3)^2 = 9$, so $2(9) - 3(4)$
3. Multiply: $18 - 12$
4. Subtract: $6$

If you had written $2 \cdot -3^2$ without parentheses, you might square only the $3$ and get $-9$. Parentheses keep the sign attached to the number.

Worked example 3

Find $\frac{x + y}{2}$ when $x = 11$ and $y = -3$.

$\frac{11 + (-3)}{2} = \frac{8}{2} = 4$

The fraction bar acts like parentheses: finish the top before you divide.

Evaluate and simplify. 🎯 Work each one on scratch paper first.

Name the parts. 🔽 Use the equation $5n - 6 = 19$.

Plug in. ✍️ Let $a = 5$ and $b = -2$.

1) $3a + b$

2) $ab$

3) $2(a + b)$

Key Takeaways — Part 1

• MK on the CAT-ASVAB: 15 questions, 31 minutes, about 2 minutes each, four choices, no calculator
• A number next to a letter means multiply: $3x = 3 \cdot x$
• Like terms share the same variable part; combine their coefficients
• Distribute to every term inside the parentheses, signs included: $-2(x - 5) = -2x + 10$
• When you substitute, put the number in parentheses, then follow the order of operations

Up Next: Part 2 — solving linear equations step by step.

⚖️ Linear Equations and Inequalities

Part 2 of 7 — Solving Linear Equations

The Balance Idea

An equation is a balance: the left side and the right side weigh the same. You can do anything you like to one side as long as you do the same thing to the other side. The goal is to get the variable alone.

To get $x$ alone, undo what was done to it, in reverse order:

Usually you undo addition and subtraction first, then multiplication and division.

The standard order of moves

1. Distribute to clear any parentheses.
2. Combine like terms on each side.
3. Move the variable terms to one side and the numbers to the other.
4. Divide by the coefficient.
5. Check by substituting into the original equation.

Worked Examples

Two steps

Solve $5x - 8 = 27$.

$5x - 8 + 8 = 27 + 8 \quad\Rightarrow\quad 5x = 35 \quad\Rightarrow\quad x = 7$

Check: $5(7) - 8 = 35 - 8 = 27$ ✓

Dividing by a number

Solve $\frac{x}{4} + 3 = 10$.

Subtract $3$: $\frac{x}{4} = 7$. Multiply both sides by $4$: $x = 28$.

Parentheses and variables on both sides

Solve $3(x - 2) = 2x + 5$.

1. Distribute: $3x - 6 = 2x + 5$
2. Subtract $2x$ from both sides: $x - 6 = 5$
3. Add $6$: $x = 11$

Check: $3(11 - 2) = 27$ and $2(11) + 5 = 27$ ✓

Negative coefficients

Solve $7 - 2x = 4x - 11$.

Add $2x$ to both sides: $7 = 6x - 11$. Add $11$: $18 = 6x$. Divide: $x = 3$.

Tip: move the variable to the side where its coefficient stays positive. Fewer negatives, fewer mistakes.

Fractions: clear them first

Solve $\frac{x}{2} + \frac{x}{3} = 10$.

Multiply every term by $6$, the smallest number both denominators go into:

$6 \cdot \frac{x}{2} + 6 \cdot \frac{x}{3} = 6 \cdot 10 \quad\Rightarrow\quad 3x + 2x = 60 \quad\Rightarrow\quad 5x = 60 \quad\Rightarrow\quad x = 12$

Solving a Formula for One Letter

Some MK questions hand you a formula and ask you to solve it for a different letter. Treat every other letter as if it were a number and use the same moves.

Example 1

Solve $A = \frac{1}{2}bh$ for $h$.

Multiply both sides by $2$: $2A = bh$. Divide both sides by $b$:

$h = \frac{2A}{b}$

Example 2

Solve $P = 2l + 2w$ for $w$.

Subtract $2l$: $P - 2l = 2w$. Divide by $2$:

$w = \frac{P - 2l}{2}$

A quick way to check a formula answer: pick easy numbers. If $l = 3$ and $w = 4$, then $P = 14$, and $\frac{14 - 6}{2} = 4$ ✓

Solve it. 🎯

Follow the steps. 🔽 Solve $5(x - 3) = 2x + 9$.

Your turn. ✍️ Solve each equation for $x$.

1) $9 - 3x = 2x - 16$

2) $\frac{x}{5} - 2 = 4$

3) $\frac{x}{3} + \frac{x}{6} = 7$

Key Takeaways — Part 2

• Whatever you do to one side, do to the other side
• Order of moves: distribute → combine → move variables → divide → check
• Undo addition and subtraction before multiplication and division
• Clear fractions by multiplying every term by a common denominator
• To solve a formula for a letter, treat the other letters as numbers
• Always check by substituting into the original equation: on the ASVAB it takes seconds and catches sign slips

Up Next: Part 3 — turning sentences into equations.

⚖️ Linear Equations and Inequalities

Part 3 of 7 — Turning Words into Equations

Translate, Then Solve

Many MK questions are sentences: "Five less than three times a number is 19. What is the number?" The math is easy once the sentence becomes an equation. The skill is the translation.

Let a letter (often $n$ or $x$) stand for the number you're looking for, then translate phrase by phrase:

The "less than" switch

"$5$ less than a number" means you start with the number and take $5$ away: $n - 5$, not $5 - n$. The same goes for "$5$ more than a number": $n + 5$ (order doesn't matter for addition, but it does for subtraction).

"$5$ less than a number" → $n - 5$ "A number less than $5$" → $5 - n$

Consecutive integers

Even and odd both step by $2$. Whether they come out even or odd depends on the starting number $n$.

Worked Examples

A number sentence

"Five less than three times a number is $19$."

• three times a number: $3n$
• five less than that: $3n - 5$
• is $19$: $3n - 5 = 19$

Solve: $3n = 24$, so $n = 8$. Check: $3(8) - 5 = 19$ ✓

Consecutive integers

"The sum of three consecutive integers is $72$. What are they?"

$n + (n + 1) + (n + 2) = 72 \quad\Rightarrow\quad 3n + 3 = 72 \quad\Rightarrow\quad 3n = 69 \quad\Rightarrow\quad n = 23$

The integers are $23$, $24$ and $25$. Read the question again: if it asks for the largest, the answer is $25$, not $23$.

Two people, one total

"Maria is $4$ years older than twice her brother's age. Together their ages add to $31$. How old is her brother?"

Let $b$ = the brother's age. Then Maria is $2b + 4$.

$b + (2b + 4) = 31 \quad\Rightarrow\quad 3b + 4 = 31 \quad\Rightarrow\quad 3b = 27 \quad\Rightarrow\quad b = 9$

Her brother is $9$ and Maria is $22$. Check: $9 + 22 = 31$ ✓

A fee plus a rate

"A storage unit costs a 40-dollar move-in fee plus 55 dollars a month. Dev has paid 425 dollars in total. How many months has he rented it?"

$40 + 55m = 425 \quad\Rightarrow\quad 55m = 385 \quad\Rightarrow\quad m = 7$

The one-time fee is added once; the monthly charge is multiplied by the number of months.

Translate the sentence. 🎯

Phrase to math. 🔽 Choose the expression that matches each phrase.

Write it, solve it. ✍️

1) Three times a number, decreased by $8$, is $22$. What is the number?

2) The sum of three consecutive integers is $57$. What is the smallest of the three?

3) Jordan is $3$ times as old as his niece. Their ages add up to $48$. How old is the niece?

Key Takeaways — Part 3

• Name the unknown with a letter, then translate phrase by phrase
• "is" means $=$; "of" and "times" mean multiply
• "$5$ less than a number" is $n - 5$: the order switches
• Consecutive integers: $n, n + 1, n + 2$; consecutive even or odd: $n, n + 2, n + 4$
• A one-time fee is added once; a per-month or per-item charge multiplies
• Re-read what the question asks for (smallest, largest, the other person's age) before choosing

Up Next: Part 4 — inequalities, and the one rule that makes them different from equations.

⚖️ Linear Equations and Inequalities

Part 4 of 7 — Linear Inequalities

What an Inequality Says

An equation like $x = 4$ has one answer. An inequality like $x > 4$ has a whole range of answers: $5$, $4.1$, $100$, and every other number bigger than $4$.

Solve it like an equation, with one exception

You solve an inequality with the same moves as an equation: add, subtract, multiply and divide both sides. One rule is new:

When you multiply or divide both sides by a negative number, flip the inequality sign.

Why? Start with $2 < 5$, which is true. Multiply both sides by $-1$: $-2$ and $-5$. Now $-2$ is the bigger number, so the true statement is $-2 > -5$. The sign had to flip.

Adding or subtracting a negative number does not flip the sign. Only multiplying or dividing by one does.

Graphing on a Number Line

• An open circle means the endpoint is not included: use it for $<$ and $>$.
• A closed (filled) dot means the endpoint is included: use it for $\le$ and $\ge$.
• Shade toward the numbers that work. For "greater than," that is to the right; for "less than," to the left.

A quick check: pick a number in the shaded part and test it in the inequality. In Figure 1, $3$ is shaded, and $3 > 2$ is true ✓

Worked Examples

No flip needed

Solve $3x - 7 < 11$.

$3x < 18 \quad\Rightarrow\quad x < 6$

We divided by positive $3$, so the sign stays.

Dividing by a negative

Solve $8 - 4x \le 20$.

1. Subtract $8$: $-4x \le 12$
2. Divide by $-4$ and flip: $x \ge -3$

Check with a number that should work, like $x = 0$: $8 - 0 = 8 \le 20$ ✓. And one that shouldn't, like $x = -5$: $8 + 20 = 28$, which is not $\le 20$ ✓

A compound inequality

Solve $-4 \le 2x + 2 < 10$.

Do each move to all three parts:

$-6 \le 2x < 8 \quad\Rightarrow\quad -3 \le x < 4$

The graph is a segment from $-3$ (closed dot) to $4$ (open circle).

A word problem

"A truck can carry at most $2{,}000$ pounds. It already holds $650$ pounds. Each crate weighs $90$ pounds. What is the greatest number of crates it can take?"

$650 + 90c \le 2000 \quad\Rightarrow\quad 90c \le 1350 \quad\Rightarrow\quad c \le 15$

The greatest number is $15$ crates. Forgetting the $650$ pounds already on board gives $2000 \div 90 \approx 22$, which would overload the truck.

Solve the inequality. 🎯

Read the graph. 🎯 Use Figure 3.

Match the graph to the inequality. 🎯 Use Figure 4.

Words to symbols. 🔽 Let $s$ be a score.

Finish the solution. ✍️

1) $4x - 3 \ge 25$, so $x \ge$ ?

2) $-6x < 30$, so $x >$ ?

3) How many integers satisfy $-2 < x \le 3$?

Key Takeaways — Part 4

• Solve inequalities with the same moves as equations
• Multiply or divide by a negative → flip the sign. Adding or subtracting never flips it
• Open circle: $<$ or $>$ (endpoint left out). Closed dot: $\le$ or $\ge$ (endpoint included)
• "At least" is $\ge$; "at most" and "no more than" are $\le$
• In a compound inequality, do every move to all three parts
• Test one number from your answer in the original inequality

Up Next: Part 5 — solving two equations at the same time.

⚖️ Linear Equations and Inequalities

Part 5 of 7 — Systems of Two Equations

Two Unknowns, Two Clues

A system is two equations that must both be true at the same time:

$x + y = 10 \qquad x - y = 4$

A solution is a pair of numbers, written $(x, y)$, that works in both equations. Here the solution is $(7, 3)$: $7 + 3 = 10$ ✓ and $7 - 3 = 4$ ✓. The pair $(6, 4)$ works in the first equation but not the second, so it is not a solution.

On the ASVAB you can often check the answer choices by plugging each pair into both equations. When you need to solve from scratch, use one of two methods.

Method 1: Substitution

Use it when one equation already says "$y =$" or "$x =$".

1. Take the expression for one letter.
2. Substitute it into the other equation. Now there is only one letter.
3. Solve, then plug back in to find the second letter.

Method 2: Elimination

Use it when the same letter has matching or opposite coefficients.

1. Add or subtract the equations so one letter cancels.
2. Solve for the letter that's left.
3. Plug back in to find the other one.

If no letter cancels yet, multiply one equation by a number first so that it will.

Worked Examples

Substitution

Solve: $y = 2x - 1$ and $3x + y = 14$.

Replace $y$ in the second equation with $2x - 1$:

$3x + (2x - 1) = 14 \quad\Rightarrow\quad 5x - 1 = 14 \quad\Rightarrow\quad 5x = 15 \quad\Rightarrow\quad x = 3$

Then $y = 2(3) - 1 = 5$. Solution: $(3, 5)$. Check in the second equation: $9 + 5 = 14$ ✓

Elimination by adding

Solve: $x + y = 10$ and $x - y = 4$.

The $y$-terms are opposites, so add the equations:

$(x + y) + (x - y) = 10 + 4 \quad\Rightarrow\quad 2x = 14 \quad\Rightarrow\quad x = 7$

Then $7 + y = 10$, so $y = 3$.

Multiply first, then eliminate

Solve: $2x + 3y = 12$ and $x - y = 1$.

Multiply the second equation by $3$ so the $y$-terms become opposites: $3x - 3y = 3$. Add it to the first:

$5x = 15 \quad\Rightarrow\quad x = 3$

Then $3 - y = 1$, so $y = 2$. Solution: $(3, 2)$.

A word problem

"A school play sold $40$ tickets. Adult tickets cost 8 dollars and child tickets cost 5 dollars. Ticket sales totaled 260 dollars. How many adult tickets were sold?"

Let $a$ = adult tickets and $c$ = child tickets.

$a + c = 40 \qquad 8a + 5c = 260$

From the first equation, $c = 40 - a$. Substitute:

$8a + 5(40 - a) = 260 \quad\Rightarrow\quad 3a + 200 = 260 \quad\Rightarrow\quad a = 20$

So $20$ adult tickets (and $20$ child tickets). Check: $160 + 100 = 260$ ✓

Rare case: if both letters cancel and you are left with something false, like $0 = 5$, the system has no solution. If you are left with something always true, like $0 = 0$, it has infinitely many.

Solve the system. 🎯

Eliminate. 🔽 Solve $4x + y = 11$ and $2x + y = 7$.

Two systems. ✍️

The system $x = y + 3$ and $2x + y = 18$:

1) $x$ = ?

2) $y$ = ?

3) A unit orders $25$ meals, some chicken and some vegetarian. There are $7$ more chicken meals than vegetarian meals. How many vegetarian meals are there?

Key Takeaways — Part 5

• A solution of a system is a pair $(x, y)$ that works in both equations
• Fastest check on the test: plug each answer choice into both equations
• Substitution: when one equation already says $y = \ldots$ or $x = \ldots$
• Elimination: add or subtract so one letter cancels; multiply first if you need to
• After finding one letter, plug back in for the other, then answer the letter the question asks for

Up Next: Part 6 — test strategy and the traps the answer choices are built from.

⚖️ Linear Equations and Inequalities

Part 6 of 7 — Test Strategy and Common Traps

Your Clock on MK

On the computer version, a few things change how you should work:

• You can't go back. Once you answer, the question is gone. Check your work before you click.
• It adapts to you. A right answer usually brings a harder question next. Don't panic if questions get tougher: that's usually a good sign.
• Every question might count. Some questions may be unscored tryouts, and you can't tell which, so treat every one as scored.
• Don't rush-guess at the end. If time runs out, unfinished questions are scored as random guesses. The official advice is to keep reading and genuinely answering when time is short, because the computer test applies a relatively large penalty for a string of wrong answers at the end of a subtest.

Two minutes is plenty for one equation. Use it: solve, then check.

Four Moves That Save Points

1. Check by substituting

Plug your answer back into the original equation. If both sides don't match, you made a slip somewhere. This takes about 15 seconds and catches most sign errors.

2. Work backward from the choices

When an equation looks messy, try the answer choices. Start with a middle value: if it's too big, you know to try smaller ones.

Example: "Which value of $x$ makes $3(x - 2) + x = 18$ true?" Try $6$: $3(4) + 6 = 18$ ✓. Done, without solving anything.

3. Answer the question that was asked

Read the last line again before you choose. Many questions ask for something other than $x$:

"If $3x + 4 = 19$, what is $6x + 1$?"

$3x = 15$, so $x = 5$, and $6x + 1 = 31$. The choices will include $5$ (the value of $x$) and $15$ (the value of $3x$) to catch people who stop early. A shortcut: $6x$ is double $3x$, so $6x = 30$ without ever finding $x$.

4. Know where the wrong choices come from

Test writers build wrong answers from common mistakes. If you know the mistakes, you can spot the traps.

Spot the trap. 🎯 Each wrong choice comes from a real mistake. Find the right one.

Find the error. 🔽 A student solved $3 - 4(x - 2) = 23$ like this:

Answer what's asked. ✍️

1) If $5x + 3 = 28$, what is $10x - 1$?

2) If $2y - 7 = 9$, what is $y + 10$?

Key Takeaways — Part 6

• About 2 minutes per MK question on the CAT-ASVAB: enough to solve and check
• You can't go back, so check by substituting before you answer
• Stuck? Plug in the answer choices, starting from a middle value
• Re-read the last line: the question may ask for $2x$, $x + 4$ or $y$
• Wrong choices come from predictable mistakes: half-distributed negatives, unflipped inequalities, reversed "less than," stopping early
• If time is short, keep genuinely working: a run of wild guesses at the end costs more than it saves

Up Next: Part 7 — the mixed mastery set.

⚖️ Linear Equations and Inequalities

Part 7 of 7 — Mixed Mastery Set

How to Use This Set

Twelve questions from every part of this lesson, mixed the way the ASVAB mixes them: a few quick ones, several medium ones and a few that take more steps.

• Give yourself about 2 minutes per question, the CAT-ASVAB pace for MK.
• Use scratch paper, no calculator.
• Check each answer before you move on, just as you would on the real test where you can't go back.

Set A — warm up. 🎯

Set B — the core. 🎯

Set C — more steps. 🎯

Finish strong. ✍️ Solve for $x$. Fractions are fine.

1) $4 - (x + 5) = 2x$

2) $\frac{x}{4} - 1 = \frac{x}{6}$

Key Takeaways — Linear Equations and Inequalities

• Substitute with parentheses; distribute to every term, signs included
• Solve with the same move on both sides: distribute → combine → move → divide → check
• Translate words phrase by phrase; "less than" switches the order
• Inequalities solve like equations, except the sign flips when you multiply or divide by a negative
• Systems: substitution when a letter is already alone, elimination when coefficients match
• On the CAT-ASVAB you have about 2 minutes per MK question and no going back: solve, check, answer what was asked

Next step: take the exit quiz for this topic. Anything you miss sends you back to the part that teaches it.