Exercises: Solve linear equations and inequalities - Common Core Math - HS Algebra

A

Warm-Up: Review What You Know

These problems review skills from earlier courses.

1.

Which value of $x$ satisfies $2x + 6 = 14$ ?

A.

$x = 3$

B.

$x = 4$

C.

$x = 10$

D.

$x = 5$

2.

Which graph correctly represents the solution set $x > -2$ ?

A.

Closed circle at $-2$ , arrow pointing right

B.

Open circle at $-2$ , arrow pointing left

C.

Open circle at $-2$ , arrow pointing right

D.

Closed circle at $-2$ , arrow pointing left

3.

A student solves $3x = 12$ for $x$ by dividing both sides by 3. If instead the equation is $ax = 12$ (where $a \neq 0$ ), which expression gives $x$ ?

A.

$x = 12 - a$

B.

$x = 12a$

C.

$x = a + 12$

D.

$x = \frac{12}{a}$

B

Fluency Practice

Solve each equation or inequality. Show your steps.

C

Varied Practice

Problems use different formats. Read each one carefully.

D

Word Problems

Set up and solve an equation or inequality for each problem.

1.

A phone plan charges a flat fee of 25 dollars per month plus 10 cents per text message. A second plan charges a flat fee of 10 dollars per month plus 20 cents per text message.

For how many text messages per month do the two plans cost exactly the same? (Let $x$ be the number of messages and solve.)

2.

A student earns 12 dollars per hour babysitting. She wants to save at least 180 dollars this month.

What is the minimum number of whole hours she must work? (Set up and solve an inequality for $h$ , the number of hours.)

3.

Ohm's Law relates voltage $V$ (volts), current $I$ (amperes), and resistance $R$ (ohms): $V = IR$ .

1.

Solve $V = IR$ for $R$ . Type your answer as a formula in the form "R = V/I".

2.

A circuit has voltage $V = 24$ volts and resistance $R = 6$ ohms. Use your formula from part (a) to find the current $I$ in amperes.

4.

A club charges an annual membership fee of 40 dollars plus 8 dollars per class. A non-member pays 15 dollars per class with no fee.

What is the minimum number of whole classes for which membership costs less than paying as a non-member?

E

Error Analysis

Each problem shows a student's work containing an error. Identify the mistake.

1.

Marcus solved $-5x > 30$ :

Divide both sides by $-5$ : $x > -6$ .
Solution: $x > -6$ .

What error did Marcus make, and what is the correct solution?

A.

Marcus computed $30 \div (-5) = -6$ incorrectly.

B.

Marcus should have subtracted 5 from both sides, not divided.

C.

Marcus forgot to flip the inequality symbol when dividing by a negative number; the correct solution is $x < -6$ .

D.

Marcus used the wrong operation; he should have multiplied both sides by $-5$ .

2.

Priya solved $x - 3 > 7$ :

She noted: "I am subtracting (adding a negative), so the inequality flips."
She wrote: $x < 10$ .
Solution: $x < 10$ .

Which rule did Priya misapply, and what is the correct solution?

A.

Priya should have added 3 to the right side only.

B.

Priya flipped the inequality when she should not have. Adding or subtracting never changes the inequality direction. The correct solution is $x > 10$ .

C.

Priya made an arithmetic error; $7 + 3 = 11$ , not 10.

D.

Priya should have divided both sides by $-3$ .

F

Challenge / Extension

These problems require multi-step reasoning or deeper conceptual understanding.

1.

Solve for $x$ : $\quad ax - b = cx + d$ , where $a \neq c$ .

Express your answer as a fraction. Type your answer in the form "(d+b)/(a-c)".

2.

A student claims: "When I solve an inequality and get a statement like $5 > 10$ , I must have made an algebra mistake — that statement is false."