Algebra 1 Basics for Beginners

UltimateAlgebra・2 minutes read

To solve equations, follow the steps to isolate the variable by performing opposite operations, handling different types of equations like multi-step or absolute value, solving inequalities, graphing on number lines, and applying the concept to word problems. Functions must have each input value mapping to a single output to be considered a function, as opposed to relations where one input can have multiple outputs.

Insights

  • Solving equations involves performing opposite operations to isolate the variable; for one-step equations, remove terms by performing inverse operations, while multi-step equations require systematically eliminating all attached terms to find the solution.
  • When dealing with absolute value equations, consider both positive and negative solutions by solving two separate equations; for radical equations, square both sides to remove the radical term and solve for x.

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Recent questions

  • How do you solve one-step equations?

    By moving variable terms to one side.

  • What is the process for solving multi-step equations?

    Eliminate terms sequentially, then isolate the variable.

  • How do you solve absolute value equations?

    Solve for positive and negative values separately.

  • What is the approach to solving radical equations?

    Isolate the radical term and square both sides.

  • How do you solve inequalities?

    Treat them like equations, reversing signs when needed.

Related videos

Summary

00:00

Equation Solving Techniques for Various Scenarios

  • To solve one-step equations, move all terms with the variable to one side by performing the opposite operation; for example, in x + 2 = 5, subtract 2 from both sides to get x = 3.
  • For two-step equations like 2x + 3 = 11, first eliminate addition/subtraction terms, then tackle multiplication/division; subtract 3 from both sides, then divide by 2 to find x = 4.
  • Multi-step equations, such as 3x^2 + 8 = 20, require removing all terms attached to the variable; subtract 8, divide by 3, then find the square root to get x = 2.
  • When x appears twice in an equation, like 4x + 5 = 9 + 2x, move one set of x terms to one side by subtracting 2x; then, eliminate other terms to find x = 2.
  • Absolute value equations, e.g., |x + 3| = 7, involve solving for both positive and negative values; solve two equations, x + 3 = 7 and x + 3 = -7, to find x = 4 and x = -10.
  • In equations like |x + 1| + 6 = 9, isolate the absolute value term by removing other terms; solve for x by equating the absolute value to positive and negative values, finding x = -4 or x = 2.
  • Radical equations, where x is under a square root, require isolating the radical term; square both sides to eliminate the square root and solve for x.
  • Rational equations, like 4 / (x - 5) = 3 / x, involve cross-multiplying to remove fractions; simplify to find x = -5.
  • To solve for x in y = mx + b, rearrange the equation by moving terms to isolate x; subtract b, then divide by m to find x = (y - b) / m.
  • In inequalities, such as -3x + 1 > 7, treat them like equations but remember to reverse the inequality sign when dividing or multiplying by a negative; solve to find x < -2.
  • Combined inequalities, like -3 < x + 8 < 20, require solving each part separately and then combining the solutions; subtract 8 to find -11 < x < 12.
  • Graphing inequalities involves plotting points on a number line and shading appropriately; for x > -4, plot -4 with an open circle and draw an arrow to the right.
  • In word problems, like packaging 2,500 gallons into 20 boxes with 100 gallons left, identify the total, the group (boxes), and the remaining amount; solve by setting up equations to find the number of gallons in each box.

17:54

Solving Equations and Identifying Functions Efficiently

  • To find the value of X, start with the equation 20x = 2,400 by subtracting 100 from both sides, then divide by 20 to get X = 120, indicating there were 120 gallons in each box.
  • For faster problem-solving, skip unnecessary steps and directly write the two-step equation, such as 20x + 100 = 2,500, then solve by subtracting 100 and dividing by 20 to get the answer.
  • In identifying functions, ensure each input value corresponds to only one output value; a relation is not a function if an input has multiple outputs, as seen in the example where input two has outputs of seven and five, making it not a function.
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