MATH 130 · FRIDAY, SEPTEMBER 4

Equations with Fractions and Roots

Clear denominators, rearrange formulas, and turn factors into solutions

50 minutes · Quiz 2 on Wednesday, September 9 at 2:10 PM

What the class data says

Review R.6

61%

Rebuild rational-expression skills where they support equation solving.

Sections 1.1, 1.2, 1.4

42%

Prioritize fractional equations, special cases, formulas, and roots.

Today’s pathway: restrictions → clear fractions → classify solutions → rearrange → solve rational and quadratic equations.

Focus · 3 minFriday, Sep 4

Entrance check: prerequisites

Work in your notes · 3 minutes
  1. State the restriction: $\dfrac{x+2}{x-3}$.
  2. Simplify: $\dfrac{6x^2-12x}{3x}$.
  3. Solve: $\dfrac{x}{2}+3=7$.

Retrieve · 5 minFriday, Sep 4

Clear fractions before solving

Solve and check

$\dfrac{x+1}{3}-\dfrac{x-2}{4}=2$

  1. LCD $=12$. Multiply every term by $12$.
  2. $4(x+1)-3(x-2)=24$
  3. $4x+4-3x+6=24$, so $x=14$.
  4. Check: $15/3-12/4=5-3=2$.

Keep each numerator grouped until after the LCD is distributed.

Model · 6 minFriday, Sep 4

One, none, or infinitely many

True statement remains

$2(x-3)=2x-6$

$2x-6=2x-6$

Infinitely many solutions

False statement remains

$\dfrac{x+1}{2}=\dfrac{x+5}{2}$

$x+1=x+5$

No solution

If the variable remains, solve for one value. If it disappears, classify the statement.

Explain · 5 minFriday, Sep 4

Rearrange a formula with fractions

Solve for b

$d=\dfrac{a+b}{c}$

  1. Restriction: $c\ne0$.
  2. Clear the denominator: $cd=a+b$.
  3. Isolate the requested variable: $b=cd-a$.
  4. Check by substituting $cd-a$ for $b$.

Treat letters like numbers. Undo operations in reverse order and preserve grouping.

Model · 5 minFriday, Sep 4

Solve a rational equation safely

Solve and verify

$\dfrac{2}{x-1}=\dfrac{3}{x+2}$

  1. Restrictions: $x\ne1,-2$.
  2. Multiply by $(x-1)(x+2)$: $2(x+2)=3(x-1)$.
  3. $2x+4=3x-3$, so $x=7$.
  4. $7$ is allowed, and both sides equal $1/3$.

Write restrictions first. A solution that makes an original denominator zero must be rejected.

Model · 5 minFriday, Sep 4

From factoring to quadratic roots

Solve by factoring

$6x^2+x-2=0$

  1. $ac=-12$. Use $4$ and $-3$: $6x^2+4x-3x-2=0$.
  2. Group: $2x(3x+2)-1(3x+2)=0$.
  3. Factor: $(2x-1)(3x+2)=0$.
  4. Zero-product property: $x=\dfrac12$ or $x=-\dfrac23$.

Factoring produces expressions. Setting each factor equal to zero produces the roots.

Connect · 6 minFriday, Sep 4

Mixed guided practice

Work first · compare methods · then reveal
  1. Solve: $\dfrac{2x-1}{3}+\dfrac{x+2}{6}=\dfrac52$.
  2. Solve $v=\dfrac{d-d_0}{t}$ for $d$.
  3. Solve: $4x^2-12x=0$.

Practice · 6 minFriday, Sep 4

Quiz 2 readiness check

No notes · 4 minutes
Show the decisive step
  1. Solve: $\dfrac{x}{3}-\dfrac{x-2}{5}=\dfrac4{15}$.
  2. Classify: $\dfrac{x+4}{2}=\dfrac{x-2}{2}+3$.
  3. Solve $V=\dfrac13Bh$ for $h$.
  4. Solve: $3x^2-7x+2=0$.

Assess · 6 minFriday, Sep 4

Choose the next ALEKS target

If fractions caused the miss

Start with fractional coefficients, binomial denominators, or rational equations.

If factoring caused the miss

Start with roots of $ax^2+bx=0$ or a quadratic with $a>1$.

Quiz 2: Wednesday, September 9 at 2:10 PM. Continue ALEKS Review R.6 and Sections 1.1, 1.2, and 1.4.

Target · 2 minFriday, Sep 4