MATH 130 · Review R.1–R.5

Algebra Foundations: Factor First

Sets, intervals, absolute value, exponent rules, factoring, and rational expressions

50–55 minutes · Offline capable · Quiz 1 · Sep 2 at 2:10 PM

Learning roadmap and pacing

  1. 1

    translate intervals and solve basic absolute-value equations

  2. 2

    simplify products and quotients using exponent rules

  3. 3

    factor completely by choosing a method in a reliable order

  4. 4

    simplify rational expressions while preserving restrictions

Canvas pacing anchor

Quiz 1 · Sep 2 at 2:10 PM

ALEKS Review R.1–R.3 and R.4–R.5 due Sep 2 at 2:10 PM

Coverage note

This deck is the sequential study path. Official quiz and test coverage can be adjusted in Canvas or in class.

Activate · 2 minUnit 1 sequence

Entrance retrieval

3 minutes · no notes
  1. Write $-2<x\le4$ in interval notation.
  2. Simplify $x^7/x^{10}$ with positive exponents.
  3. Factor $6x^3-9x^2$ completely.

Commit to all three before comparing.

Activate · 4 minUnit 1 sequence

Intervals and absolute value describe location

Interval endpoints

$x\ge a\Rightarrow[a,\infty)$

$x<b\Rightarrow(-\infty,b)$

Distance from zero

$|u|=c\Rightarrow u=c$ or $u=-c$, when $c\ge0$.

$|u|$ cannot be negative.

Language check

Union combines regions; intersection keeps only their overlap. Infinity always receives a parenthesis.

Explain · 5 minUnit 1 sequence

Exponent rules: name the operation first

×

Same base product: add exponents.

÷

Same base quotient: subtract exponents, top minus bottom.

^

Power of a power: multiply exponents.

$a^{-n}=1/a^n$ for $a\ne0$. A negative exponent changes position, not sign.

Explain · 5 minUnit 1 sequence

Factoring: use the decision order

1

Take out the GCF.

2

Count terms: 2, 3, or 4.

3

Use a pattern, trinomial method, or grouping.

4

Multiply back to verify.

Explain · 5 minUnit 1 sequence

Rational expressions inherit factoring

Simplify $\dfrac{3x-27}{x-8}\cdot\dfrac{5x-40}{6x-54}$.

  1. Factor every polynomial.
  2. Cancel common factors, not terms: $\dfrac{3(x-9)}{x-8}\cdot\dfrac{5(x-8)}{6(x-9)}$.
  3. Result: $5/2$, with $x\ne8,9$ from the original expression.

Model · 6 minUnit 1 sequence

Error analysis: what is invalid?

Student claim

$\dfrac{x+3}{x}=3$ after “canceling x.”

Diagnose

Cancellation applies to factors. Since $x+3$ is a sum, no $x$ factor can cancel. A quick check at $x=1$ gives $4\ne3$.

Feedback · 4 minUnit 1 sequence

You try: connect the foundations

Work first · then reveal

  1. Write $(-\infty,5]\cap[1,8)$ as one interval.
  2. Solve $|x-2|=5$.
  3. Simplify $\dfrac{m^2n^{-1}}{m^{-4}n^3}$.
  4. Factor $2x^3-18x$ completely.

Practice · 5 minUnit 1 sequence

Quiz 1 checkpoint: readiness check

Quiz 1 checkpoint
Closed notes · show complete work
  1. Translate $x<-1$ or $x\ge3$ into interval notation.
  2. Solve $|2x-1|=7$.
  3. Simplify $(2x^{-2}y^3)^2$ with positive exponents.
  4. Factor $3x^2-12x+12$ completely.
  5. Simplify $\dfrac{x^2-9}{x^2+x-6}$ and state restrictions.

Assess · 8 minUnit 1 sequence

Exit ticket and next step

Before you leave
  1. Name one skill you can now complete without notes.
  2. Name one error pattern to watch.
  3. Schedule the next short practice before the checkpoint.
Next step

Next: Unit 1B — linear equations, formulas, and lines

Synthesize · 3 minUnit 1 sequence