Grade 10 MATH Term 1 Week 6 - 7: Solving Absolute Value Equations & Inequalities |MATATAG Q1 Tagalog

Main ideas / lessons

1) Absolute value basics (meaning and notation)

• Absolute value represents the distance from 0 on a number line.
• Distance is never negative, so absolute value results are non-negative.
• Notation uses two vertical bars: (|\;\cdot\;|)

Examples

• (|3| = 3)
• (|-3| = 3)
• (|0| = 0)

Key terms

• Argument: everything inside the absolute value bars
  • Example: in (|x-8|), the argument is (x-8)
• Constant (P): the value that the absolute value equals
  • Example: in (|x-8| = 12), (P = 12)

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2) Solving absolute value equations (one variable)

General approach: isolate the absolute value, then apply the correct rule.

Rule 1 (when (P>0))

If (|\text{argument}| = P), then:

• (\text{argument} = P)
• (\text{argument} = -P)

Solve both and combine solutions into a set.

Rule 2 (when (P=0))

If (|\text{argument}| = 0), then:

• (\text{argument} = 0)

This gives only one solution.

Rule 3 (when (P<0))

If (|\text{argument}| = -P) (meaning the right side is negative), then:

• No solution

The answer can be written as the null set.

Special note / exemption mentioned (quadratic absolute value equation)

For a quadratic absolute value equation (treated as an exception), the lesson suggests:

• Set the squared factor to zero
• Use factoring

Example idea shown:

• Factor into terms like ((x+3)(x+4)=0)
• Then (x=-3) and (x=-4)

Solution representation (equations)

• Listing/roster form: (S={-4, 20}) (order can vary)
• Set-builder form: ({x \mid x=-4 \text{ or } x=20})

The video mainly emphasizes set representations; roster and set-builder are explicitly demonstrated.

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3) Solving absolute value inequalities (one variable)

The video teaches 4 rules based on whether the inequality is:

• less than / less than or equal
• greater than / greater than or equal

It also stresses number line/graphing logic using open vs closed circles:

• “<” and “>” → open circles (endpoints not included)
• “≤” and “≥” → closed circles (endpoints included)

Rule 1: (|x| < P)

If (P>0):

• (-P < x < P)

Rule 2: (|x| \le P)

If (P>0):

• (-P \le x \le P)

Rule 3: (|x| > P)

If (P>0):

• (x < -P) or (x > P)

(Intervals outside the middle region.)

Rule 4: (|x| \ge P)

If (P>0):

• (x \le -P) or (x \ge P)

(Intervals outside the middle region, endpoints included.)

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4) How to write solutions for inequalities (multiple notations)

Common templates given:

• Inequality form Example: (-6 < x < 12) (use < vs ≤ exactly as shown)

• Interval notation

  • Parentheses for exclusive endpoints: ((-6, 12))
  • Brackets for inclusive endpoints: ([-6, 12])
  • For “or” solutions, use union: ((-\infty,-2] \cup [7,\infty))

• Set-builder form

  • Example: (x \in \mathbb{R}) such that (-6 < x < 12)

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Methodology / step-by-step instruction bullets (as presented)

Solving absolute value equations

• Step 1: Isolate the absolute value on one side.
• Step 2: Use the appropriate rule:
  • If (|\text{argument}| = P) with (P>0):
    • Set argument = (P)
    • Set argument = (-P)
    • Solve both
  • If (|\text{argument}| = 0):
    • Set argument = 0
    • Solve once
  • If (|\text{argument}|) equals a negative number:
    • No solution (null set)
• Step 3: Express the final answers as:
  • roster/listing set and/or set-builder notation

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Solving absolute value inequalities

• Step 1: Isolate the absolute value (transform using inequality properties as needed).
• Step 2: Convert using the matching rule:
  • (|x|<P) → (-P<x<P)
  • (|x|\le P) → (-P\le x\le P)
  • (|x|>P) → (x<-P) or (x>P)
  • (|x|\ge P) → (x\le -P) or (x\ge P)
• Step 3: Solve the resulting simple inequalities (usually linear).
• Step 4 (graphing/number line):
  • Mark endpoints:
    • open circle for strict inequalities (<, >)
    • closed circle for inclusive inequalities (≤, ≥)
  • Shade:
    • between endpoints for “<” or “≤”
    • outside endpoints for “>” or “≥”
• Step 5: Write the solution in one or more forms:
  • inequality notation, interval notation (with parentheses/brackets), and/or set-builder notation
  • Use union when the solution is “or” (outside intervals)

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Speakers / sources featured

• Primary speaker: An instructor/teacher (Tagalog voiceover teaching math on the “Mat YouTube channel”)
• Channel/source: MATATAG Q1 Tagalog / “Mat YouTube channel” (as named in the video title and opening announcement)