Notes
Solving linear equations
- Use inverse operations on both sides. Distribute and combine like terms first.
- With variables on both sides, collect them on one side. A false result means no solution; a true result means infinitely many solutions.
Literal equations
- Solve for a chosen variable by treating the others as numbers: $\displaystyle A = P + Prt \Rightarrow t = \frac{A - P}{Pr}$.
Inequalities
- Solve like equations, but reverse the sign when you multiply or divide by a negative number.
- Compound inequalities: "and" gives an overlap like $1 \leq x < 4$; "or" gives two separate pieces.
Modeling
- Define a variable, write an equation or inequality, solve, and interpret with units. Words like "at most" mean $\leq$ and "at least" mean $\geq$.
Topics in this unit
Each topic has its own short notes and practice.
Practice
$3x = 15$, so $x = 5$.
Subtract $P$: $A - P = Prt$. Then divide by $Pr$.
$-2x < -6$. Dividing by $-2$ reverses the sign: $x > 3$.
Add 2 to all parts: $3 \leq 3x < 12$. Divide by 3: $1 \leq x < 4$.
$3 + 2m \leq 25$ gives $m \leq 11$.
$-2x \geq 8$ gives $x \leq -4$. Only $-5$ satisfies this.
Free response
Write your answer first, then open the scoring guide.
Free-response question
Maya is saving for a bike that costs $240. She already has $60 and saves $15 each week.
(a) Write an expression for the total she has after $w$ weeks, and find the total after 8 weeks.
Show the answer and scoring guide
Total $= 60 + 15w$. After 8 weeks: $60 + 15(8) = 180$, so she has $180.
Check off the points you earned:
(b) Write and solve an inequality for the number of weeks until she can afford the bike.
Show the answer and scoring guide
$60 + 15w \geq 240 \Rightarrow 15w \geq 180 \Rightarrow w \geq 12$. She needs at least 12 weeks.
Check off the points you earned:
(c) Solve $T = 60 + 15w$ for $w$, then find $w$ when $T = 330$.
Show the answer and scoring guide
$\displaystyle w = \frac{T - 60}{15}$. For $T = 330$: $\displaystyle w = \frac{270}{15} = 18$ weeks.
Check off the points you earned: