Solve Linear Equations and Inequalities

In this lesson:

• Solve equations with numbers AND with letter coefficients
• Apply the critical sign-flip rule for inequalities
• Represent solution sets three ways: inequality, interval, number line

Learning Objectives for This Lesson

By the end of this lesson, you will:

1. Solve linear equations — numeric and literal
2. Apply the sign-flip rule for inequalities
3. Represent solutions in inequality, interval, and number line forms
4. Identify special cases: no solution and infinitely many solutions

From Numbers to Letters: Same Steps

• → one answer
• for → a formula (handles every specific case)

Literal equations use letters where numbers usually appear. The steps are the same.

The 5-Step Algorithm for Linear Equations

Any linear equation can be solved with these steps:

1. Distribute — clear all parentheses
2. Combine like terms on each side
3. Move variable terms to one side
4. Move constant terms to the other side
5. Divide by the coefficient

Works for numeric and literal equations alike.

Worked Example: Solving a Numeric Equation

Solve:

1. Distribute →
2. Collect variables →
3. Divide → ✓

Literal Equations: Same Steps, Letters as Constants

Solve for :

1. Collect:
2. Factor:
3. Divide:

Same process as numeric — just letters.

Applied Literal Equations: Ohm's Law

Formula: (Voltage = Current × Resistance)

Solve for : Treat and as constants

This is formula rearrangement — the same skill as literal equations, applied to science.

Guided Practice: Rearranging a Literal Equation

Solve for .

1. Move -terms left:
2. Move -terms right
3. Factor out
4. Divide by the coefficient of

Work through all steps, then advance

Practice Problems: Solving Linear Equations

Solve each:

3. Solve for
4. Solve for

Pause and solve all four before advancing

Answers to Linear Equations Practice

The Critical Rule for Inequalities

Equality properties carry over — with one exception:

Multiplying or dividing by a NEGATIVE reverses the inequality symbol.

Why the Sign Flips: Visual Proof

Example: is true. Multiply both sides by :

Without reversing: is false.

Worked Example 1: No Flip Needed

Solve:

Step 1: Subtract 5:

Step 2: Multiply by 3 (positive):

Number line: open circle at 9, arrow pointing right

Interval notation:

Worked Example 2: Flip Required

Solve:

Divide both sides by (negative!):

Number line: closed circle at , arrow pointing right

Interval notation:

Worked Example 3: Multi-Step Inequality

Solve:

Distribute:

Move terms:

Number line: closed circle at 12, left arrow

Interval:

Check-In: Identify When to Flip

Solve . Does the symbol flip? Why or why not?

Work through the solution:

• What operation isolates ?
• Does that operation require flipping?
• What is the solution in inequality notation?

Verify: does satisfy the original inequality?

Three Representations of Inequality Solutions

Reading the Solution Set Reference Chart

Key rules: parenthesis open circle strict inequality; bracket closed circle ≤ or ≥

Parentheses vs. Brackets: The Rule

• Strict ( or ): endpoint excluded → open circle → parenthesis (
• Non-strict ( or ): endpoint included → closed circle → bracket [
• Infinity: always parenthesis — never reached

After a flip: apply the rule to the new symbol.

Guided Practice: Writing Inequality Representations

Express in all three forms.

• Inequality notation: already given —
• Interval notation: — what goes in the bracket?
• Number line: circle at — open or closed?

Which arrow direction? Left or right?

Practice Problems: Converting Inequality Representations

Convert each to all three forms:

4. Solution: all satisfying

Pause and complete all four before advancing

Answers to Inequality Representation Practice

1. → → open circle, right
2. → → closed circle, left
3. → → closed at , open at 3
4. → → open circle, right

Special Cases: When the Variable Disappears

If the variable drops out, the remaining statement determines the answer:

• False (e.g., ) → No solution
• True (e.g., ) → All real numbers

This is not an error — it IS the answer.

Special Case: No Solution Worked Example

Solve:

Distribute:

Subtract :

This is always false — no value of makes this true.

Solution: No solution (empty set, )

All Real Numbers: Worked Example

Solve:

Distribute:

Subtract :

This is always true — every value of satisfies this.

Solution: All real numbers,

Check-In: Recognizing Special Case Equations

Solve .

Work through the steps. When the variable disappears:

• Is the remaining statement true or false?
• What is the solution set?

Express the answer using interval notation.

Mixed Practice: Equations and Inequalities

Solve each. Note any special cases.

2. Solve for

Complete all five, then advance

Answers to Mixed Practice Problems

1. (flip! ÷ ) →
2. (literal)
3. — always true → all real numbers
4. →

Key Takeaways from This Lesson

• Equations: 5-step algorithm — same for numeric and literal
• Literal: treat other letters as constants; factor out the target
• Inequalities: multiply/divide by negative → FLIP
• Non-strict → bracket; strict → parenthesis; → parenthesis
• Variable drops out: false → no solution; true → all reals

What's Next: Systems of Equations

Next lesson: HSA.REI.C.6 — Systems of Linear Equations

Two equations, two unknowns — solved by graphing, substitution, or elimination.

The single-variable skills you practiced today are essential building blocks for systems.