One-Variable Equations and Inequalities

1. Substitute $x = 7$ into $3x - 4 = 17$:

$3(7) - 4 = 21 - 4 = 17$

Since $17 = 17$, $x = 7$ is a solution.

2. Let $n$ = the unknown number.

"Multiplied by 6, then 5 subtracted, giving 31":

$6n - 5 = 31$

Add $5$ to both sides:

$6n = 36$

Divide both sides by $6$:

$n = 6$

Check: $6(6) - 5 = 36 - 5 = 31$. Correct.

3. Subtract $3$ from both sides:

$\frac{y}{4} = 6$

Multiply both sides by $4$:

$y = 24$

Check: $\frac{24}{4} + 3 = 6 + 3 = 9$. Correct.

4. Let $c$ = number of cards Jordan started with.

After giving away $14$, she has $23$:

$c - 14 = 23$

Add $14$ to both sides:

$c = 37$

Jordan started with 37 cards.

Check: $37 - 14 = 23$. Correct.

5. "Greater than $-2$" means $x > -2$.

"At most $5$" means $x \leq 5$.

Combined inequality:

$-2 < x \leq 5$

Number line description: Open circle at $-2$ (not included), closed circle at $5$ (included), shaded region between them.

6. $2x + 1 \leq 13$

Subtract $1$ from both sides:

$2x \leq 12$

Divide both sides by $2$:

$x \leq 6$

Solution set: All real numbers less than or equal to $6$, written $x \leq 6$ or $(-\infty, 6]$.

Number line: Closed circle at $6$, shaded to the left.