Praxis Algebra I (5162)

A linear equation is an equation in which the variable appears only to the first power. To solve, isolate the variable using inverse operations.

For addition/subtraction equations, subtract or add the same value to both sides. Example: x + 7 = 12. Subtract 7: x = 5.

For multiplication/division equations, divide or multiply both sides by the same nonzero value. Example: 4x = 20. Divide by 4: x = 5.

Two-step equations require two inverse operations. Generally, undo addition or subtraction first, then undo multiplication or division.

Multi-step equations may require distributing, combining like terms, and moving variable terms to one side.

When variables appear on both sides, collect variable terms on one side and constants on the other.

Identity: An equation that is true for all values of the variable. When solving, you obtain a statement like 0 = 0 or 5 = 5. The solution set is all real numbers.

Contradiction: An equation with no solution. When solving, you obtain a false statement like 0 = 5.

A literal equation contains multiple variables. Solving for a specific variable means isolating that variable on one side.

The absolute value of a number is its distance from zero on the number line, always nonnegative. For |x| = a where a > 0, the solutions are x = a or x = −a.

If the absolute value equals a negative number, there is no solution. If it equals zero, there is exactly one solution.

An inequality compares expressions using <, >, ≤, or ≥. Solving inequalities follows the same steps as equations with one critical exception: when multiplying or dividing by a negative number, reverse the inequality sign.

On a number line, use an open circle for < or > (endpoint not included) and a closed circle for ≤ or ≥ (endpoint included). Shade in the direction of the solution set.

Conjunction (AND): Both conditions must be satisfied. Written as a < x < b or as two inequalities joined by "and." The solution is the intersection of the two sets.

Disjunction (OR): At least one condition must be satisfied. The solution is the union of the two sets.

For |x| < a (where a > 0), the solution is −a < x < a. This is a conjunction.

For |x| > a (where a > 0), the solution is x < −a or x > a. This is a disjunction.