GED Algebra Practice

Algebraic problem solving is roughly 55 percent of the GED math test, which makes it the highest-return topic on the exam. These questions cover the specific skills the test asks for.

Question 1 of 22

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Solve the system: x + y = 12 and y = x + 4. What is y?

Algebra study guide

Algebra is the larger half of GED Mathematical Reasoning. The good news is that the test returns to the same handful of skills again and again. This guide walks through each one with a worked example, in the order that makes the next one easier.

Solving one-variable linear equations

A linear equation is solved by undoing operations in reverse order. Whatever was done to the variable last is undone first. In 4x + 9 = 37, the x was multiplied by 4 and then had 9 added, so you subtract 9 first and divide by 4 second.

Every step must be applied to both sides of the equals sign. That rule is the whole method. If parentheses are involved, distribute before moving terms. If the variable appears on both sides, collect the variable terms on one side and the plain numbers on the other.

Worked example

Solve 3(x − 4) = x + 10.

  1. Distribute the 3: 3x − 12 = x + 10.
  2. Subtract x from both sides: 2x − 12 = 10.
  3. Add 12 to both sides: 2x = 22.
  4. Divide by 2: x = 11.
  5. Check: 3(11 − 4) = 21 and 11 + 10 = 21. Both sides match.

Turning words into equations

Many GED algebra items describe a situation in words and ask which equation or expression models it. Translate one phrase at a time. “Is” becomes an equals sign. “Of” after a fraction or percent means multiply. “Less than” reverses the order: 5 less than a number is n − 5, not 5 − n.

Name the unknown before you write anything. Writing “let h = hours worked” keeps you from mixing up which quantity the variable stands for, which is the root of most modeling errors.

Worked example

A plumber charges a $65 visit fee plus $48 per hour. A bill came to $257. How many hours did the job take?

  1. Let h = hours worked. The bill is 65 + 48h.
  2. Set it equal to the total: 65 + 48h = 257.
  3. Subtract 65: 48h = 192.
  4. Divide by 48: h = 4 hours.

Inequalities

Inequalities are solved exactly like equations with one extra rule: when you multiply or divide both sides by a negative number, the inequality sign flips. Forgetting that flip is one of the most common ways to lose a point on this topic.

Word phrases map to symbols in a fixed way. “At least” means greater than or equal to (≥). “At most” and “no more than” mean less than or equal to (≤). “More than” and “fewer than” are strict (> and <). On a number line, a closed dot means the endpoint is included and an open dot means it is not.

Worked example

Solve 7 − 2x > 15.

  1. Subtract 7 from both sides: −2x > 8.
  2. Divide both sides by −2 and flip the sign: x < −4.
  3. Check with x = −5: 7 − 2(−5) = 17, and 17 > 15 is true.

Slope, intercepts, and graphs of lines

Slope measures how much y changes for each one-unit change in x. From two points, slope = (y₂ − y₁) ÷ (x₂ − x₁). A positive slope rises from left to right, a negative slope falls, a horizontal line has slope 0, and a vertical line has an undefined slope.

In slope-intercept form, y = mx + b, the number m is the slope and b is the y-intercept, the point where the line crosses the y-axis. In word problems, b is usually a starting amount or flat fee, and m is the rate. Parallel lines have equal slopes. Perpendicular lines have slopes that are negative reciprocals, such as 2 and −1/2.

Worked example

Find the equation of the line through (1, 5) and (3, 11).

  1. Slope: (11 − 5) ÷ (3 − 1) = 6 ÷ 2 = 3.
  2. Substitute one point into y = 3x + b: 5 = 3(1) + b, so b = 2.
  3. The line is y = 3x + 2.
  4. Check the other point: 3(3) + 2 = 11. Correct.

Function notation and evaluating expressions

f(x) is read “f of x” and simply names a rule. f(4) means replace every x in the rule with 4 and compute. The most frequent error is with negative inputs: put the negative number in parentheses before you square it, because (−3)² is 9 while −3² is −9.

A table or graph can also define a function. The test may ask whether a relationship is a function: it is a function if every input has exactly one output. On a graph, any vertical line crosses a function at most once.

Worked example

If g(x) = x² − 4x + 1, find g(−3).

  1. Substitute with parentheses: (−3)² − 4(−3) + 1.
  2. Compute each term: 9 + 12 + 1.
  3. g(−3) = 22.

Systems of equations and simple quadratics

A system asks for the values that make two equations true at once. Substitution works well when one equation already isolates a variable. Elimination works well when the coefficients line up so adding or subtracting the equations cancels a variable. On a multiple-choice question, plugging each answer choice into both equations is often fastest.

Quadratics on the GED are usually solved by factoring. Write the equation equal to zero, factor it into two binomials, and set each factor equal to zero. The quadratic formula is on the formula sheet if factoring is not obvious.

Worked example

Solve x² − x − 12 = 0.

  1. Find two numbers that multiply to −12 and add to −1: −4 and 3.
  2. Factor: (x − 4)(x + 3) = 0.
  3. Set each factor to zero: x = 4 or x = −3.

Exponent rules and polynomials

Exponent rules are not on the formula sheet, so it pays to know them by heart. When you multiply powers with the same base, add the exponents: x⁴ · x³ = x⁷. When you divide, subtract them: x⁸ ÷ x² = x⁶. When you raise a power to a power, multiply them: (x³)² = x⁶. Any nonzero number to the power 0 is 1, and a negative exponent means a reciprocal, so x⁻² = 1/x².

A common mistake is to apply these rules to the numbers in front. In (3x²)³, the 3 is cubed too, giving 27x⁶, not 3x⁶ or 9x⁶. Another is to add exponents when terms are added: x² + x³ cannot be combined, because they are not like terms.

Polynomials are sums of terms such as 4x² − 3x + 7. To add or subtract them, combine like terms, the ones with the same variable and the same exponent. When subtracting, put the second polynomial in parentheses and change the sign of every term inside it before you combine. To multiply two binomials, multiply each term in the first by each term in the second, then combine the middle terms. Two products are worth spotting on sight: (a + b)(a − b) = a² − b², and (a + b)² = a² + 2ab + b². Forgetting the middle term 2ab is the most frequent error when squaring a binomial.

Worked example

Simplify (2x² − 5x + 4) − (x² − 3x − 6).

  1. Change the sign of every term in the second polynomial: 2x² − 5x + 4 − x² + 3x + 6.
  2. Combine the x² terms: 2x² − x² = x².
  3. Combine the x terms: −5x + 3x = −2x.
  4. Combine the constants: 4 + 6 = 10.
  5. The result is x² − 2x + 10.
  6. Check with x = 1: the original gives 1 − (−8) = 9, and 1 − 2 + 10 = 9. They match.

Checklist before test day

If you can do each item below without looking anything up, you are ready for the algebra share of the test.

  • Solve an equation with parentheses and variables on both sides.
  • Write an equation from a word problem with a flat fee and a rate.
  • Flip the inequality sign when dividing by a negative.
  • Find slope from two points and write y = mx + b.
  • Evaluate a function at a negative input without a sign error.
  • Factor a simple trinomial and solve a two-equation system.

Algebra practice set 1: 22 questions with answers

Work each question on paper first, then open the answer to check your reasoning.

  1. 1.Solve the system: x + y = 12 and y = x + 4. What is y?

    1. A.6
    2. B.16
    3. C.8
    4. D.4
    Show answer

    Answer: C. 8

    Substitute x + 4 for y: x + x + 4 = 12, so 2x = 8 and x = 4. Then y = 4 + 4 = 8; choice 4 is the value of x.

  1. 2.Solve the system: 3x + 2y = 16 and 5x − 2y = 0. What is x?

    1. A.x = 2
    2. B.x = 8
    3. C.x = 5
    4. D.x = 3
    Show answer

    Answer: A. x = 2

    Adding the equations cancels the y-terms: 8x = 16, so x = 2 (and y = 5). Choice x = 5 is the value of y.

  1. 3.A coffee shop sells small drinks for $3 and large drinks for $5. One morning it sold 60 drinks for a total of $236. How many large drinks did it sell?

    1. A.32
    2. B.28
    3. C.47
    4. D.40
    Show answer

    Answer: B. 28

    Replace s with 60 − l: 3(60 − l) + 5l = 236, so 180 + 2l = 236 and l = 28. Choice 32 is the number of small drinks.

  1. 4.How many solutions does the system y = 3x + 2 and 6x − 2y = 8 have?

    1. A.infinitely many solutions
    2. B.one solution
    3. C.two solutions
    4. D.no solutions
    Show answer

    Answer: D. no solutions

    Solving the second equation for y gives y = 3x − 4. Both lines have slope 3 but different y-intercepts, so they are parallel and never meet. Infinitely many solutions would require the same line.

  1. 5.Factor: x² − 10x + 21

    1. A.(x + 3)(x − 7)
    2. B.(x − 3)(x − 7)
    3. C.(x − 1)(x − 21)
    4. D.(x + 3)(x + 7)
    Show answer

    Answer: B. (x − 3)(x − 7)

    The numbers must multiply to +21 and add to −10, so both are negative: −3 and −7. (x + 3)(x + 7) has the right product but gives +10x.

  1. 6.Factor completely: 3x² + 12x

    1. A.3(x² + 4x)
    2. B.3x(x + 4)
    3. C.x(3x + 4)
    4. D.3x(x + 12)
    Show answer

    Answer: B. 3x(x + 4)

    Both terms share 3 and x, so the greatest common factor is 3x, leaving 3x(x + 4). 3(x² + 4x) is equivalent but not complete, because x can still be factored out.

  1. 7.What are the solutions of x² − 16 = 0?

    1. A.x = 4 or x = −4
    2. B.x = 4 only
    3. C.x = 16 or x = −16
    4. D.x = 8 or x = −8
    Show answer

    Answer: A. x = 4 or x = −4

    Factor as (x − 4)(x + 4) = 0, so x = 4 or x = −4. Choosing 4 only misses that (−4)² is also 16.

  1. 8.The product of two consecutive positive integers is 132. What is the smaller integer?

    1. A.66
    2. B.10
    3. C.11
    4. D.12
    Show answer

    Answer: C. 11

    n(n + 1) = 132 gives n² + n − 132 = 0, which factors as (n + 12)(n − 11) = 0, so n = 11. Choice 12 is the larger integer.

  1. 9.The table shows a function. x: 1, 2, 3, 4 y: 5, 8, 11, 14 Which equation fits every pair in the table?

    1. A.y = 3x + 5
    2. B.y = 5x
    3. C.y = 3x + 2
    4. D.y = x + 4
    Show answer

    Answer: C. y = 3x + 2

    y rises by 3 each time x rises by 1, so the slope is 3, and 5 = 3(1) + b gives b = 2. y = x + 4 matches only the first pair.

  1. 10.A line graph of a generator’s fuel tank passes through (0, 1,200) and (6, 0), where x is hours of use and y is gallons of fuel. What does the slope of the line tell you?

    1. A.The tank loses 200 gallons each hour.
    2. B.The tank loses 1,200 gallons each hour.
    3. C.The tank holds 1,200 gallons when full.
    4. D.The tank empties after 200 hours.
    Show answer

    Answer: A. The tank loses 200 gallons each hour.

    The slope is (0 − 1,200) ÷ (6 − 0) = −200, so the fuel drops 200 gallons per hour. The 1,200 value is the y-intercept, the starting amount, not the rate.

  1. 11.If g(x) = 5 − 2x, for what value of x is g(x) = −7?

    1. A.x = 19
    2. B.x = 6
    3. C.x = 1
    4. D.x = −6
    Show answer

    Answer: B. x = 6

    Set 5 − 2x = −7, so −2x = −12 and x = 6. Choice 19 is g(−7), which uses −7 as the input instead of the output.

  1. 12.Simplify: (x⁴)³

    1. A.x⁷
    2. B.x¹²
    3. C.x⁶⁴
    4. D.3x⁴
    Show answer

    Answer: B. x¹²

    A power raised to a power multiplies the exponents: 4 × 3 = 12. Choice x⁷ adds the exponents, which is the rule for multiplying two powers.

  1. 13.Which expression is equivalent to (2x³)(5x⁻¹)?

    1. A.7x²
    2. B.10x⁻³
    3. C.10x⁴
    4. D.10x²
    Show answer

    Answer: D. 10x²

    Multiply the coefficients (2 × 5 = 10) and add the exponents: 3 + (−1) = 2. Choice 7x² adds the coefficients instead of multiplying them.

  1. 14.An event hall holds no more than 250 people. 38 staff members will be working the event. Which inequality shows the number of guests g who can attend?

    1. A.38g ≤ 250
    2. B.g + 38 ≥ 250
    3. C.g + 38 ≤ 250
    4. D.g − 38 ≤ 250
    Show answer

    Answer: C. g + 38 ≤ 250

    Guests plus staff cannot go over 250, so g + 38 ≤ 250, which means g ≤ 212. The ≥ sign would require at least 250 people in the hall.

  1. 15.A freelancer charges $45 per hour. She wants to earn at least $1,350 this week and has already earned $360 from an earlier job. What is the minimum number of hours she must still work?

    1. A.38
    2. B.30
    3. C.22
    4. D.8
    Show answer

    Answer: C. 22

    She still needs 1,350 − 360 = $990, and 45h ≥ 990 gives h ≥ 22. Choice 30 ignores the $360 she already earned.

  1. 16.Solve: −3(x − 4) ≥ 2x − 8

    1. A.x ≥ 4
    2. B.x ≤ −4
    3. C.x ≤ 4
    4. D.x ≥ 0.8
    Show answer

    Answer: C. x ≤ 4

    Distribute to get −3x + 12 ≥ 2x − 8, then −5x ≥ −20. Dividing by −5 flips the sign, so x ≤ 4. Keeping the ≥ sign is the most common mistake here.

  1. 17.An electrician’s bill is a flat fee plus an hourly rate. A 3-hour job cost $230 and a 5-hour job cost $370. What is the flat fee?

    1. A.$70
    2. B.$76.67
    3. C.$160
    4. D.$20
    Show answer

    Answer: D. $20

    Two extra hours added $140, so the rate is $70 per hour. Then 230 − 3(70) = $20 is the flat fee. Choice $70 is the hourly rate, not the fee.

  1. 18.A gym’s total cost is modeled by C = 35m + 50, where m is the number of months. What does the 50 represent?

    1. A.the monthly fee
    2. B.the cost after one month
    3. C.the number of months
    4. D.the one-time start fee
    Show answer

    Answer: D. the one-time start fee

    The 50 is added once no matter how many months pass, so it is a one-time fee, the cost when m = 0. The 35 is the monthly fee because it is multiplied by m.

  1. 19.A parking garage charges $4 for the first hour and $2.50 for each additional hour. Which expression gives the cost for h hours, where h is at least 1?

    1. A.4 + 2.5(h − 1)
    2. B.4 + 2.5h
    3. C.6.5h
    4. D.2.5 + 4(h − 1)
    Show answer

    Answer: A. 4 + 2.5(h − 1)

    The first hour costs $4, and each hour after it adds $2.50, so there are h − 1 additional hours. Choice 4 + 2.5h charges $2.50 for the first hour on top of the $4.

  1. 20.A number decreased by 7 is equal to one third of the number. What is the number?

    1. A.3.5
    2. B.21
    3. C.−3.5
    4. D.10.5
    Show answer

    Answer: D. 10.5

    Write x − 7 = x/3. Multiply by 3 to get 3x − 21 = x, so 2x = 21 and x = 10.5. Choice 21 forgets to divide by 2 at the end.

  1. 21.Ana’s age is 4 less than three times Ben’s age. The sum of their ages is 56. How old is Ana?

    1. A.15
    2. B.45
    3. C.38
    4. D.41
    Show answer

    Answer: D. 41

    Write Ana’s age as 3b − 4. Then b + (3b − 4) = 56 gives 4b = 60 and b = 15, so Ana is 3(15) − 4 = 41. Choice 15 is Ben’s age.

  1. 22.A line passes through (3, −1) and has a slope of −2. What is its y-intercept?

    1. A.5
    2. B.−7
    3. C.1
    4. D.−5
    Show answer

    Answer: A. 5

    Substitute into y = −2x + b: −1 = −2(3) + b, so −1 = −6 + b and b = 5. Choice −7 subtracts 6 instead of adding it when solving for b.

More practice sets

Algebra has 4 practice sets. None of these questions appear in the full practice tests.

What GED algebra questions ask you to do

GED algebra questions cluster around a short list of tasks: solve a linear equation for a variable, write an equation that models a described situation, find slope from two points or from an equation, solve a small system, evaluate a function at a given input, and translate phrases like at least or no more than into inequality symbols.

Quadratics appear, but lightly, usually factoring a difference of squares or a simple trinomial rather than deriving anything. If you can confidently isolate a variable and read slope-intercept form, you have covered most of what is scored.

Where test takers lose points on GED algebra

Sign errors are the single most common cause of a wrong answer, especially when distributing a negative across parentheses or moving a term across the equals sign. Work the step, then check it by substituting your answer back into the original equation. On a multiple-choice test, substitution is often faster than solving forward.

The second common loss is misreading which quantity the question wants. A problem may walk you through finding x and then ask for the total cost, which is 25 plus x. Underline the final question before you start computing.

A GED algebra study order that works

Get one-variable equations to the point of boredom first. Then move to slope and linear graphs, since those show up both in algebra questions and in data-analysis questions about trends. Systems and function notation come last. They appear less often, and they are much easier once the basics are automatic.

Frequently asked questions

How much algebra is on the GED math test?
Algebraic problem solving accounts for about 55 percent of the Mathematical Reasoning test.
Do I need to know the quadratic formula?
It is provided on the formula sheet. Most quadratic questions on the GED can be solved by factoring or by testing the answer choices.