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InequalitiesAlgebraHigh schoolabout 1:35

Solve 2x^2 + 5x - 3 > 0: animated lesson

A narrated, step-by-step inequalities video lesson with captions. Final answer: (-∞, -3) ∪ (1/2, ∞).

Video with a synthetic English voice and captions. Pick another language for subtitles in that language, or open the interactive version.

Video narrated by a synthetic voice (Piper TTS, en_US-ljspeech-high voice (LJ Speech dataset, public domain)); the steps come from our maths engine and the answer was checked independently. Download MP4 (0.6 MB) · Captions (WebVTT)

Steps on screen

  1. 1. The problem
    Worked example
    2x2+5x−3>02{x}^{2}+5x-3 > 0
  2. 2. Steps, part 1
    1. (2x−1)(x+3)>0(2x - 1)(x + 3) > 0
      Move everything to one side and factor
    2. Find the critical points
    3. Test one point in each interval (sign chart)
    4. Decide the endpoints
  3. 3. Steps, part 2
    1. x∈(−∞,−3)∪(12,∞)x \in (-\infty, -3) \cup (\frac{1}{2}, \infty)
      Write the solution set
  4. 4. The answer
    (−∞,−3)∪(12,∞)(-\infty, -3) \cup (\frac{1}{2}, \infty)

    x ∈ (-∞, -3) ∪ (1/2, ∞)

  5. 5. Check
    • x = -4: satisfies the inequality.
    • x = -1.25: does not satisfy the inequality.
    • x = 2: satisfies the inequality.

Transcript

The problem. Here is the problem: 2x squared plus 5x minus 3 greater than 0. Solve the inequality and give the answer in interval notation.

Steps, part 1. Move everything to one side and factor. A sign chart needs the form f(x) compared with 0, written as a product/quotient of factors. Find the critical points. The sign can only change at negative 3 (zero), 1 over 2 (zero). Test one point in each interval (sign chart). (-∞, negative 3): test x equals negative 4 → sign plus → included; (negative 3, 1 over 2): test x equals negative 1.25 → sign minus → excluded; (1 over 2, ∞): test x equals 2 → sign plus → included. Decide the endpoints. negative 3: excluded; 1 over 2: excluded.

Steps, part 2. Write the solution set.

The answer. So the answer is (-∞, negative 3) ∪ (1 over 2, ∞). The inequality holds exactly for x in (-∞, negative 3) ∪ (1 over 2, ∞). Points where the expression is undefined are never included, and endpoints are included only for less than or equal to or greater than or equal to where the expression equals zero.

Check. x equals negative 4: satisfies the inequality. x equals negative 1.25: does not satisfy the inequality. x equals 2: satisfies the inequality.

About inequalities

Solve linear and quadratic inequalities and write the answer as an interval.

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Solve 2x^2 + 5x - 3 > 0: Inequalities Step-by-Step Lesson