Free printable summer review

Algebra 1 Summer Review

A free 40-problem Algebra 1 summer review PDF with typeset equations, tables, coordinate and geometry diagrams, modeling practice, and a complete answer key for students entering Geometry or Algebra 2.

40 problems8 skill areasAbout 100 minutesAnswer key included
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What this review measures

Algebra 1 provides the symbolic language used in every later high-school math course. This expanded packet balances procedural fluency with interpretation: students solve and represent equations, connect tables and coordinate diagrams to linear and quadratic functions, manipulate polynomial and radical expressions, and build models from realistic constraints.

For parents and educators

Students should show one algebra step per line and label the quantities in every modeling problem. A calculator is useful for checking arithmetic after a section, but exact fractions and radicals should remain exact unless a question requests a decimal.

Preview the complete review

Every question below appears in the printable packet. Students should show their work and simplify answers unless directed otherwise.

Equations and inequalities

Solve and represent linear equations and inequalities.

  1. 1Solve 3(2x−5)−4=2(x+7)\displaystyle 3(2x-5)-4=2(x+7).
  2. 2Solve the compound inequality −5≤2x+1<9\displaystyle -5\leq2x+1<9.
  3. 3Solve ∣3x−2∣=10\displaystyle |3x-2|=10.
  4. 4Solve x−14+x+32=5\displaystyle \frac{x-1}{4}+\frac{x+3}{2}=5.
  5. 5The area of a trapezoid is A=12h(b1+b2)\displaystyle A=\frac{1}{2}h(b_1+b_2). Solve for h\displaystyle h.bases b1 and b2; height h
  6. 6Solve −3(2x−5)>9\displaystyle -3(2x-5)>9.

Linear functions

Move among equations, points, tables, graphs, and rate interpretations.

  1. 7Write the line through (−1,6)\displaystyle (-1,6) and (3,−2)\displaystyle (3,-2) in slope-intercept form.
  2. 8Find the x\displaystyle x- and y\displaystyle y-intercepts of 3x−4y=12\displaystyle 3x-4y=12.
  3. 9For f(x)=−4x+9\displaystyle f(x)=-4x+9, solve f(x)=−15\displaystyle f(x)=-15.
  4. 10Use the table to find the slope and write the linear rule.
    xy
    -27
    03
    3-3
  5. 11Points A and B are shown. Find the slope and equation of the line through them.AB
  6. 12Write the equation of the line parallel to y=12x−4\displaystyle y=\frac{1}{2}x-4 that passes through (6,1)\displaystyle (6,1).

Systems

Solve systems and identify the number of solutions.

  1. 13Solve 3x+2y=16\displaystyle 3x+2y=16 and x−2y=0\displaystyle x-2y=0.
  2. 14Solve 2x−3y=7\displaystyle 2x-3y=7 and 4x−6y=14\displaystyle 4x-6y=14.
  3. 15Tickets cost 8\displaystyle 8 dollars for students and 12\displaystyle 12 dollars for adults. There are 35\displaystyle 35 tickets totaling $340. How many of each?
  4. 16Solve y=x+5\displaystyle y=x+5 and 2x+y=14\displaystyle 2x+y=14.
  5. 17How many liters of 30% solution and 60% solution should be mixed to make 12 liters of 45% solution?

Exponents and polynomials

Use exponent rules and operate on polynomial expressions.

  1. 18Simplify (2x−3y2)2\displaystyle (2x^{-3}y^{2})^{2} using positive exponents.
  2. 19Multiply (x−5)(2x+3)\displaystyle (x-5)(2x+3).
  3. 20Factor 6x2−15x\displaystyle 6x^{2}-15x.
  4. 21Factor x2−x−20\displaystyle x^{2}-x-20.
  5. 22Simplify 3x2y−16x−1y3\displaystyle \frac{3x^{2}y^{-1}}{6x^{-1}y^{3}} using positive exponents.
  6. 23Subtract (x2+3x−4)−(2x2−x+5)\displaystyle (x^{2}+3x-4)-(2x^{2}-x+5).

Quadratics

Analyze quadratic forms, tables, and equations.

  1. 24Solve x2−9x+20=0\displaystyle x^{2}-9x+20=0.
  2. 25Find the vertex of y=(x+2)2−7\displaystyle y=(x+2)^{2}-7.
  3. 26Solve 2x2+x−6=0\displaystyle 2x^{2}+x-6=0.
  4. 27Rewrite x2+6x+1\displaystyle x^{2}+6x+1 in vertex form.
  5. 28Use the table to identify the zeros and vertex of the quadratic.
    xy
    06
    10
    2-2
    30
    46
  6. 29Factor and solve 3x2+2x−8=0\displaystyle 3x^{2}+2x-8=0.

Radicals and sequences

Simplify radicals and recognize arithmetic and geometric patterns.

  1. 30Simplify 72\displaystyle \sqrt{72}.
  2. 31Solve x+5=x−1\displaystyle \sqrt{x+5}=x-1. Check for extraneous solutions.
  3. 32Find the 20th term of 7,11,15,…\displaystyle 7,11,15,\ldots.
  4. 33Find the 6th term of 3,6,12,…\displaystyle 3,6,12,\ldots.
  5. 34Simplify 320−45\displaystyle 3\sqrt{20}-\sqrt{45}.

Modeling

Build functions and equations, then interpret key features.

  1. 35A ball's height is h(t)=−16t2+48t+5\displaystyle h(t)=-16t^{2}+48t+5. When does it reach maximum height?
  2. 36A taxi charges $3.50 plus $2.20 per mile. Write a function C(m)\displaystyle C(m) and find C(8)\displaystyle C(8).
  3. 37An account begins with $500 and grows 4% yearly. Write an exponential model.
  4. 38A right triangle has legs x\displaystyle x and x+7\displaystyle x+7 and hypotenuse 17\displaystyle 17. Find both leg lengths.legs x and x + 7; hypotenuse 17

Challenge

Connect algebraic and graphical meaning.

  1. 39If (x−2)\displaystyle (x-2) is a factor of x2+kx−10\displaystyle x^{2}+kx-10, find k\displaystyle k.
  2. 40A system has no solution. What must be true about the slopes and y\displaystyle y-intercepts of its two lines?
View the complete answer key

Open-ended explanations may use different wording while showing the same reasoning.

Equations and inequalities

1. x=334\displaystyle x=\frac{33}{4}, or 8.25\displaystyle 8.25

2. −3≤x<4\displaystyle -3\leq x<4

3. x=4\displaystyle x=4 or x=−83\displaystyle x=-\frac{8}{3}

4. x=5\displaystyle x=5

5. h=2Ab1+b2\displaystyle h=\frac{2A}{b_1+b_2}

6. x<1\displaystyle x<1

Linear functions

7. y=−2x+4\displaystyle y=-2x+4

8. x\displaystyle x-intercept (4,0)\displaystyle (4,0); y\displaystyle y-intercept (0,−3)\displaystyle (0,-3)

9. x=6\displaystyle x=6

10. Slope −2\displaystyle -2; y=−2x+3\displaystyle y=-2x+3

11. Slope 1\displaystyle 1; y=x+1\displaystyle y=x+1

12. y=12x−2\displaystyle y=\frac{1}{2}x-2

Systems

13. (4,2)\displaystyle (4,2)

14. Infinitely many solutions

15. 20 student tickets and 15 adult tickets

16. (3,8)\displaystyle (3,8)

17. 6 liters of each solution

Exponents and polynomials

18. 4y4x6\displaystyle \frac{4y^{4}}{x^{6}}

19. 2x2−7x−15\displaystyle 2x^{2}-7x-15

20. 3x(2x−5)\displaystyle 3x(2x-5)

21. (x−5)(x+4)\displaystyle (x-5)(x+4)

22. x32y4\displaystyle \frac{x^{3}}{2y^{4}}

23. −x2+4x−9\displaystyle -x^{2}+4x-9

Quadratics

24. x=4\displaystyle x=4 or x=5\displaystyle x=5

25. (−2,−7)\displaystyle (-2,-7)

26. x=32\displaystyle x=\frac{3}{2} or x=−2\displaystyle x=-2

27. (x+3)2−8\displaystyle (x+3)^{2}-8

28. Zeros x=1\displaystyle x=1 and x=3\displaystyle x=3; vertex (2,−2)\displaystyle (2,-2)

29. (3x−4)(x+2)=0\displaystyle (3x-4)(x+2)=0; x=43\displaystyle x=\frac{4}{3} or x=−2\displaystyle x=-2

Radicals and sequences

30. 62\displaystyle 6\sqrt{2}

31. x=4\displaystyle x=4

32. 83\displaystyle 83

33. 96\displaystyle 96

34. 35\displaystyle 3\sqrt{5}

Modeling

35. At t=1.5\displaystyle t=1.5 seconds

36. C(m)=2.2m+3.5\displaystyle C(m)=2.2m+3.5; C(8)=21.10\displaystyle C(8)=21.10, so the fare is $21.10

37. A(t)=500(1.04)t\displaystyle A(t)=500(1.04)^{t}

38. 8\displaystyle 8 and 15\displaystyle 15

Challenge

39. k=3\displaystyle k=3

40. The slopes are equal and the y\displaystyle y-intercepts are different

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