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Praxis Core · Mathematics (5733)

Praxis Core: Algebra Concepts

Importance ★★★★☆ · 18% (estimate)

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Written by AI. Only units that passed a separate AI check (re-reading facts and numbers) are published. Tell us if you find a mistake.

Overview

This chapter covers the core tools of algebra. You will write and evaluate expressions, solve linear equations and inequalities, and turn word problems into equations you can solve.

An equation stays balanced only if you do the same operation to both sides. Work step by step to get the variable alone, then check your answer by putting it back into the original equation.

Inequalities are solved almost the same way, with one important difference: if you multiply or divide both sides by a negative number, you must flip the inequality sign. In word problems, define the variable first and translate key words carefully.

Key concepts

  • Evaluating an expression

    Replace each variable with its given value, then calculate using the order of operations. Use parentheses around negative numbers when substituting.

    Memory tipPlug in with parentheses: x = −3 means (−3).

  • Solving a linear equation

    Use inverse operations to get the variable alone, doing the same thing to both sides. Undo addition and subtraction first, then multiplication and division.

    Memory tipKeep the scale balanced.

  • Solving an inequality

    Solve like an equation, but flip the inequality sign whenever you multiply or divide both sides by a negative number. Example: −2x > 6 becomes x < −3.

    Memory tipNegative multiply or divide = flip the sign.

  • Translating words to math

    Common key words: "is" means =, "of" means multiply, "more than" means add, "less than" means subtract (in reversed order: 5 less than x is x − 5).

    Memory tip"Less than" flips the order.

  • Distributive property

    Multiply the number outside the parentheses by every term inside: a(b + c) = ab + ac.

    Memory tipShare with everyone inside the parentheses.

  • Slope of a line

    Slope = (change in y) ÷ (change in x), or rise over run. In y = mx + b, m is the slope and b is where the line crosses the y-axis.

    Memory tipRise over run.

Often tested

  • Do the same operation to both sides of an equation.
  • When you multiply or divide an inequality by a negative number, flip the sign.
  • Check a solution by putting it back into the original equation.
  • "5 less than x" is written x − 5, not 5 − x.
  • Define what the variable stands for before writing an equation from a word problem.

3 sample questions

Verified practice questions from this unit.

Practice questionReviewedMathematics (5733) › Algebra★★★★☆

Question 1. Solve for x: 4x − 7 = 21

  1. ① x = 5.25
  2. ② x = 112
  3. ③ x = 3.5
  4. ④ x = 7
  5. ⑤ x = 28
▼ Show answer and explanation▲ Hide answer and explanation

Answer: ④ x = 7

Key point: Undo operations in reverse: add or subtract first, then divide.

  1. Add 7 to both sides: 4x = 28.
  2. Divide both sides by 4: x = 7.

Check: 4 × 7 − 7 = 28 − 7 = 21.

Wrong choices

  • "x = 3.5": This subtracts 7 from 21 instead of adding: 4x = 14. To undo − 7, add 7 to both sides.
  • "x = 28": This adds 7 to get 4x = 28 but forgets to divide by 4.
  • "x = 5.25": This divides 21 by 4 without first moving the 7.
  • "x = 112": This multiplies 28 by 4 instead of dividing.
Practice questionReviewedMathematics (5733) › Algebra★★★★☆

Question 2. A gym charges a $25 sign-up fee plus $15 per month. Which expression gives the total cost, in dollars, for m months?

  1. ① 15m
  2. ② 25 − 15m
  3. ③ 15 + 25m
  4. ④ 40m
  5. ⑤ 25 + 15m
▼ Show answer and explanation▲ Hide answer and explanation

Answer: ⑤ 25 + 15m

Key point: One-time cost + (rate × number of periods).

The $25 fee is paid once, and $15 is paid for each of the m months, so the total is 25 + 15m.

Check: for 2 months, 25 + 15 × 2 = $55, which is $25 + $15 + $15.

Wrong choices

  • "15 + 25m": This switches the fees. The $15 is charged each month; the $25 is paid only once.
  • "40m": This charges the sign-up fee every month.
  • "15m": This leaves out the one-time $25 sign-up fee.
  • "25 − 15m": The monthly fee adds to the cost; it is not subtracted.
Practice questionReviewedMathematics (5733) › Algebra★★★★☆

Question 3. Solve for x: 3(x − 4) = 2x + 5

  1. ① x = −17
  2. ② x = 9
  3. ③ x = 17
  4. ④ x = −7
  5. ⑤ x = 3.4
▼ Show answer and explanation▲ Hide answer and explanation

Answer: ③ x = 17

Key point: Distribute, then move x terms to one side and numbers to the other.

  1. Distribute: 3x − 12 = 2x + 5.
  2. Subtract 2x from both sides: x − 12 = 5.
  3. Add 12 to both sides: x = 17.

Check: 3(17 − 4) = 3 × 13 = 39, and 2 × 17 + 5 = 39.

Wrong choices

  • "x = 9": This multiplies only x by 3 and leaves −4, giving 3x − 4 = 2x + 5. The 3 must multiply both terms.
  • "x = −7": This subtracts 12 instead of adding it when moving it across: x = 5 − 12.
  • "x = 3.4": This adds 2x to the left side (5x = 17). To move 2x, subtract it from both sides.
  • "x = −17": This makes a sign error; checking −17 gives −63 on the left and −29 on the right.
Practice 5 questions from this unit →