Q: Solve the equation $$8(x + 1) = 6 - 2(8x + 7)$$ and classify the equation as a conditional, a contradiction, or an identity.

$$x = -\frac{2}{3}$$; \text{ conditional}$

Question 1

Solve the equation $$8(x + 1) = 6 - 2(8x + 7)$$ and classify the equation as a conditional, a contradiction, or an identity.

Accepted Answer

$$x = -\frac{2}{3}$$; 	ext{ conditional}$

Question 2

Which property allows you to add or subtract the same value from both sides of an equation without changing the solution?

Accepted Answer

Addition Property of Equations

Question 3

Solve for $$x$$: $$\frac{2x}{3} + \frac{x}{4} = \frac{5}{2}$$

Accepted Answer

$$x = 2$$

Question 4

What is the first step when solving a linear equation with fractions?

Accepted Answer

Multiply all terms by the least common denominator (LCD)

Question 5

Classify the equation $$5y - 6 = 18 + 5y.$$

Accepted Answer

Contradiction

Question 6

Solve and classify: $$6x + 2x + 5 = 8x + 2 + 3$$

Accepted Answer

Identity

Question 7

If $$a = b$$, which of the following is always true for any nonzero $$c$$?

Accepted Answer

$$ac = bc$$

Question 8

A chemist mixes two solutions. The equation $$\frac{x}{3} + \frac{x}{4} = 7$$ models the total volume. What is $$x$$?

Accepted Answer

$$x = 12$$

Question 9

Which of the following equations is an identity?

Accepted Answer

$$2(x + 1) = 2x + 2$$

Question 10

Solve for $$x$$: $$4(x - 2) = 2x + 8$$

Accepted Answer

$$x = 4$$

Question 11

A medical researcher finds that the equation $$\frac{1}{2}x + \frac{1}{3}x = 10$$ models the dosage of a drug. What is $$x$$?

Accepted Answer

$$x = 12$$

Question 12

Which step is necessary when parentheses are present in a linear equation?

Accepted Answer

Apply the distributive property

Question 13

Solve for $$x$$: $$3(x + 2) = 2x + 9$$

Accepted Answer

$$x = 3$$

Question 14

Which of the following best describes a contradiction?

Accepted Answer

An equation with no solution

Question 15

A historical event is modeled by $$2x - 5 = 3x + 1. $$What is $$x$$?

Accepted Answer

$$x = -6$$

Question 16

Which equation is a conditional equation?

Accepted Answer

$$x + 2 = 5$$

Question 17

Solve for $$x$$: $$\frac{x}{2} + \frac{x}{3} = 5$$

Accepted Answer

$$x = 6$$

Question 18

If $$8x + 5 = 8x + 5$$, what type of equation is this?

Accepted Answer

Identity

Question 19

Which of the following is NOT a valid step in solving $$2(x + 3) = 4x + 6$$?

Accepted Answer

Multiplying both sides by zero

Question 20

A real-life scenario: A car rental company charges a base fee plus a per-mile rate. The equation $$50 + 0.2x = 70$$ models the total cost. How many miles $$x$$ were driven?

Accepted Answer

$$x = 100$$

Linear Equations in One Variable: Properties, Solution Methods, and Equation Types

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