ALGEBRA EXERCISES

A monomial, also called power product, is a product of powers of variables with non-negative integer exponents

A polynomial is an expression consisting of variables, also called indeterminates, and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables

Examples of monomial: 8x; 5x^{2}; 3x^{2}y

Examples of binomial: 5x+6; 7x-3; x+2

Examples of trinomial: x^{2}+6x+5, 2x^{2}-3x+1, -5x^{2}+7x-3

5x+4x = 9x

3x+4y+5x+8y = 8x+12y

3√2+5√7+8√2+3√7 = 11√2+8√7

7x+4x^{2}+5x+9x^{2} = 13x^{2}+12x

(9x^{2}+6x+5)+(3x^{2}-5x-9) = 12x^{2}+x-4

(3x^{2}+7x-4)-(8x^{2}-5x+7) = 3x^{2}+7x-4-8x^{2}+5x-7 = -5x^{2}+12x-11

7x(x^{2}+2x-3) = 7x^{3}+14x^{2}-21x

(5x^{2})(3x^{4}-6x^{3}+5x-8) = 15x^{6}-30x^{5}+25x^{3}-40x^{2}

(3x-4)(2x+7) = 6x^{2}+21x-8x-28 = 6x^{2}+13x-28

(2x-5)(4x+7) = 8x^{2}-20x+14x-35 = 8x^{2}-6x-35

(2x-3)^{2} = (2x-3)(2x-3) = 4x^{2}-6x-6x+9 = 4x^{2}-12x+9

(5x-9)(2x^{2}-3x+4) = 10x^{3}-18x^{2}-15x^{2}+27x+20x-36 = 10x^{3}-33x^{2}+47x-36

x^{3}x^{4} = x^{7}

x^{9}/x^{4} = x^{5}

(x^{7})^{6} = x^{42}

x^{2}x^{3} = (x⋅x)(x⋅x⋅x) = x^{5}

(x^{2})^{3} = (x^{2})(x^{2})(x^{2}) = (x⋅x)(x⋅x)(x⋅x) = x^{6}

x^{5}/x^{2} = (x⋅x⋅x⋅x⋅x)/(x⋅x) = x⋅x⋅x = x^{3}

x^{4}/x^{7} = x^{-3} = (x⋅x⋅x⋅x)/(x⋅x⋅x⋅x⋅x⋅x⋅x) = 1/(x⋅x⋅x) = 1/x^{3}

(3x^{4}y^{5})(5x^{6}y^{7}) = 15x^{10}y^{12}

(8x^{3}y^{-2})(7x^{-8}y^{5}) = 56x^{-5}y^{3} = 56y^{3}/x^{5}

(24x^{7}y^{-2})/(6x^{4}y^{5}) = 2^{3}⋅3x^{3}y^{-7}/2⋅3 = 4x^{3}/y^{7}

(32x^{5}y^{-3}z^{4})/(40x^{-8}y^{-7}z^{-8}) = (2^{5}x^{13}y^{4}z^{12})/(2^{3}5) = 2^{2}x^{13}y^{4}z^{12}/5 = 4x^{13}y^{4}z^{12}/5

(3x^{3})^{2} = 3^{2}x^{6} = 9x^{6}

(5x)^{2} = 5^{2}x^{2} = 25x^{2}

(5+x)^{2} = (5+x)(5+x) = 25+5x+5x+x^{2} = x^{2}+10x+25

(4x^{2}y^{3})^{3} = 4^{3}x^{6}y^{9} = 64x^{6}y^{9}

(8x^{2}y^{5}z^{6})^{0} = 1

-2(5xy^{3})^{0} = -2⋅1 = -2

(5x^{-2}/y^{-3})(8x^{4}/y^{-5}) = (5y^{3}/x^{2})(8x^{4}y^{5}) = 40x^{2}y^{8}

((35x^{-3})/(40xy^{5}))((24x^{2}y^{2})/(42y^{-4})) = (5⋅7/2^{3}⋅5x^{4}y^{5})(2^{3}⋅3x^{2}y^{6})/(2⋅3⋅7) = y/(2x^{2})

((24xy)/(27x^{-2}))/((36x^{2}y^{-3})/(45xy^{4})) = ((2^{3}⋅3x^{3}y)/(3^{3}))((3^{2}5xy^{7})/(2^{2}3^{2}x^{2})) = (2⋅5x^{2}y^{8})/(3^{2}) = 10x^{2}y^{8}/9

(3x/5)/(7xy/9) = (3x/5)(9/(7xy)) = 27/(35y)

(7+2/x)/(5-3/y) = ((7+2/x)xy)/((5-3/y)xy) = (7xy+2y)/(5xy-3x) = (y(7x+2))/(x(5y-3))

x+4 = 9, x = 5; verify: 5+4 = 9, 9 = 9

3x+5 = 11, 3x = 6, x = 6/3, x = 2; verify: 3⋅2+5 = 11, 6+5 = 11, 11 = 11

2(x-1)+6 = 10, 2(x-1) = 4, x-1 = 2, x = 3; verify: 2(3-1)+6 = 10, 2⋅2+6 = 10, 4+6 = 10, 10 = 10

5-3(x+4) = 7+2(x-1), -3(x+4) = 2+2(x-1), -3x-12 = 2+2x-2, -3x-12 = 2x, -5x = 12, x = -12/5; verify: 5-3(-12/5+4) = 7+2(-12/5-1), -3(-12/5+4) = 2+2(-12/5-1), -3((-12+20)/5) = 2+2((-12-5)/5), -3(8/5) = 2+2(-17/5), -24/5 = 2-34/5, -24/5 = (10-34)/5, -24/5 = -24/5

(2/3)x+5 = 8, (2/3)x = 3, 2x = 9, x = 9/2; verify: (2/3)(9/2)+5 = 8, 3+5 = 8, 8 = 8

(3/4)x-1/3 = 1, 12((3/4)x-1/3) = 12, 9x-4 = 12, 9x = 16, x = 16/9; verify: (3/4)(16/9)-1/3 = 1, 4/3-1/3 = 1, 3/3 = 1, 1 = 1

(x+2)/5 = 7/8, 8(x+2) = 7⋅5, 8x+16 = 35, 8x = 19, x = 19/8; verify: ((19/8)+2))/5 = 7/8, (35/8)(1/5) = 7/8, 7/8 = 7/8

0.04x+0.15 = 0.09x-0.25, 100(0.04x+0.15) = (0.09x-0.25)100, 4x+15 = 9x-25, -5x = -40, x = 40/5, x = 8; verify: 0.04⋅8+0.15 = 0.09⋅8-0.25, 0.32+0.15 = 0.72-0.25, 0.47 = 0.47

x^{2}-25 = 0, x^{2} = 25, √x^{2} = √25, x = ±5, x = 5 or x = -5; x^{2}-25 = 0, (x+5)(x-5) = 0, x+5 = 0, x = -5, or x-5 = 0, x = 5

2x^{2}-18 = 0, 2x^{2} = 18, x^{2} = 18/2, x^{2} = 9, √x^{2} = √9, x = ±3, x = 3 or x = -3; 2x^{2}-18 = 0, 2(x^{2}-9) = 0, 2(x+3)(x-3) = 0, x+3 = 0, x = -3, or x-3 = 0, x = 3

3x^{2}-48 = 0, 3x^{2} = 48, x^{2} = 48/3, x^{2} = 16, √x^{2} = √16, x = ± 4, x = 4 or x = -4; 3x^{2}-48 = 0, 3(x^{2}-16) = 0, 3(x+4)(x-4) = 0, x+4 = 0, x = -4, or x-4 = 0, x = 4

x^{4}-81 = 0, (x^{2}+9)(x^{2}-9) = 0, (x^{2}+9)(x+3)(x-3), x+3 = 0, x = -3 or x-3 = 0, x = 3, the real solutions are x = 3 or x = -3; √-9 = ±3i

x^{2}-5x+6 = 0, (x-3)(x-2) = 0, x-3 = 0, x = 3, or x-2 = 0, x = 2

x^{2}-2x-15 = 0, (x+3)(x-5) = 0, x+3 = 0, x = -3, or x-5 = 0, x = 5

x^{2}+3x-28 = 0, (x+7)(x-4) = 0, x+7 = 0, x = -7, or x-4 = 0, x = 4

2x^{2}+3x-2 = 0, 2x^{2}+4x-x-2 = 0, 2x(x+2)-1(x+2) = 0, (x+2)(2x-1) = 0, x+2 = 0, x = -2, or 2x-1 = 0, 2x = 1, x = 1/2; 2x^{2}+3x-2 = 0, ax^{2}+bx+c = 0, x = (-b±√b^{2}-4ac)/(2a), x = (-3±√3^{2}-4⋅2⋅-2)/(2⋅2), x = (-3±√9+16)/4, x = (-3±√25)/4, x = (-3±5)/4, x = (-3+5)/4, x = 2/4 = 1/2, or x = (-3-5)/4, x = -8/4 = -2

6x^{2}+7x-3 = 0, 6x^{2}+9x-2x-3 = 0, 3x(2x+3)-1(2x+3) = 0, (2x+3)(3x-1) = 0, 2x+3 = 0, 2x = -3, x = -3/2, or 3x-1 = 0, 3x = 1, x = 1/3; 6x^{2}+7x-3 = 0, ax^{2}+bx+c = 0, x = (-b±√b^{2}-4ac)/(2a), x = (-7±√7^{2}-4⋅6⋅-3)/(2⋅6), x = (-7±√49+72)/12, x = (-7±√121)/12, x = (-7±11)/12, x = (-7+11)/12 = 4/12 = 1/3, or x = (-7-11)/12 = -18/12 = -3/2

x^{3}-4x^{2}-x+4 = 0, x^{2}(x-4)-1(x-4) = 0, (x-4)(x^{2}-1) = 0, (x-4)(x+1)(x-1), x-4 = 0, x = 4, or x+1 = 0, x = -1, or x-1 = 0, x = 1

Slope-intercept form of a linear equation: y = mx+b, m is the slope that is rise/run, b is the y-intercept that is the point 0,b

y = 2x-1; y = mx+b, m = 2, b = -1

y = (3/4)x-2; y = mx+b, m = 3/4, b = -2

Standard form of a linear equation: ax+by = c; y = 0, ax = c, x = c/a, x-intercept = (c/a,0); x = 0, by = c, y = c/b, y-intercept = (0,c/b)

2x-3y = 6; y = 0, 2x = 6, x = 6/2 = 3, x-intercept = (3,0); x = 0, -3y = 6, y = -6/3 = -2 = y-intercept = (0,-2)

Point-slope form of a linear equation: y-y_{1} = m(x-x_{1})

Find the equation of the line knowing the angular coefficient and a point: m = 2, P(1,3); y-y_{1} = m(x-x_{1}), y-3 = 2(x-1) is the solution in point-slope form; y-3 = 2(x-1), y-3 = 2x-2, y = 2x-2+3 = 2x+1, y = 2x+1 is the solution in slope-intercept form; y = 2x+1, -2x+y = 1 is the solution in standard form

Find the equation of the line passing through two points: P_{1}(2,4), P_{2}(-1,5); m = (y_{2}-y_{1})/(x_{2}-x_{1}) = (5-4)/(-1-2) = 1/-3 = -1/3, m = -1/3; y-y_{1} = m(x-x_{1}), y-4 = -1/3(x-2) is the solution in point-slope form; y-4 = -1/3(x-2), y-4 = (-1/3)x+2/3, y = (-1/3)x+2/3+4 = (-1/3)x+2/3+12/3 = (-1/3)x+14/3, y = (-1/3)x+14/3 is the solution in slope-intercept form; y = (-1/3)x+14/3, 3y = 3((-1/3)x+14/3), 3y = -x+14, x+3y = 14 is the solution in standard form

Find the equation of the line passing through the point P(1,3) and parallel to 2x-3y-5 = 0; 2x-3y-5 = 0, -3y = -2x+5, y = (-2/-3)x+5/-3 = (2/3)x-5/3, m = 2/3; y-y_{1} = m(x-x_{1}), y-3 = 2/3(x-1) is the solution in point-slope form; y-3 = 2/3(x-1), y-3 = (2/3)x-2/3, y = (2/3)x-2/3+3 = (2/3)x+7/3, y = (2/3)x+7/3 is the solution in slope-intercept form

Find the equation of the line passing through the point P(-2,1) and perpendicular to 3x+2y-7 = 0; 3x+2y-7 = 0, 2y = -3x+7, y = (-3/2)x+7/2, the slope of the perpendicular line is the negative reciprocal of the slope of the starting line, so m = 2/3; y-y_{1} = m(x-x_{1}), y-1 = (2/3)(x+2) is the solution in point-slope form; y-1 = (2/3)(x+2), y-1 = (2/3)x+4/3, y = (2/3)x+4/3+1 = (2/3)x+7/3, y = (2/3)x+7/3 is the solution in point-intercept form; point of intersection of the two perpendicular lines: y = (-3/2)x+7/2, y = (2/3)x+7/3, (-3/2)x+7/2 = (2/3)x+7/3, (-3/2)x-(2/3)x = 7/3-7/2, 6((-3/2)x-(2/3)x) = 6(7/3-7/2 ), -9x-4x = 14-21, -13x = -7, x = -7/-13 = 7/13, x = 7/13 = 0,538..., y = (2/3)x+7/3, y = (2/3)(7/13)+7/3 = 14/39+7/3 = (14+91)/39 = 105/39 = 35/13 = 2,692..., the point of intersection is P(7/13,35/13)

x > 2; (2,+∞)

x ≥ -1; [-1,+∞)

x < 4; (-∞,4)

x ≤ -2; (-∞,-2]

1 < x ≤ 4; (1,4]

x < -2 or x ≥ 3; (-∞,-2)∪[3,+∞)

x+4 > 5; x > 5-4, x > 1, (1,∞)

3x-5 < -8; 3x < -8+5, 3x < -3, x < -3/3, x < -1, (-∞,-1)

7-2x ≤ 12; -2x ≤ 12-7, -2x ≤ 5, x ≥ 5/(-2), x ≥ -5/2, [-5/2,+∞)

4-2x ≥ 3x+19; -2x-3x ≥ 19-4, -5x ≥ 15, x ≤ 15/(-5), x ≤ -3, (-∞,-3]

5-2(x-3) > 3+4(2x-1); 5-2x+6 > 3+8x-4, -2x+11 > 8x-1, -8x-2x > -1-11, -10x > -12, x < -12/-10, x < 12/10, x < 6/5, (-∞,6/5)

(3/2)x < -9; 2(3/2)x < -9⋅2, 3x < -18, x < -18/3, x < -6, (-∞,-6)

(1/3)x+4 ≥ 8; (1/3)x ≥ 8-4, (1/3)x ≥ 4, 3(1/3)x ≥ 4⋅3, x ≥ 12, [12,+∞)

(5/4)x-2 ≤ (3/2)x+1/3; 12((5/4)x-2) ≤ ((3/2)x+1/3)12, 15x-24 ≤ 18x+4, 15x-18x ≤ 4+24, -3x ≤ 28, x ≥ 28/-3, x ≥ -28/3, [-28/3,+∞)

(2/3)x+5 > (1/5)x+3/4; 60((2/3)x+5) > 60((1/5)x+3/4), 40x+300 > 12x+45, 40x-12x > 45-300, 28x > -255, x > -255/28, (-255/28,+∞)

|x| < 4; x < 4, x > -4, (-4,4)

|2x-3| ≥ 8; 2x-3 ≥ 8, 2x ≥ 8+3, 2x ≥ 11, x ≥ 11/2; 2x-3 ≤ -8, 2x ≤ -8+3, 2x ≤ -5, x ≤ -5/2; x ≥ 11/2 or x ≤ -5/2, (-∞,-5/2]∪[11/2,+∞)

5-3|4x+1| ≥ -9; -3|4x+1| ≥ -9-5, -3|4x+1| ≥ -14, |4x+1| ≤ -14/-3, |4x+1| ≤ 14/3; 4x+1 ≤ 14/3, 4x ≤ 14/3-1, 4x ≤ (14-3)/3, 4x ≤ 11/3, x ≤ 11/3⋅4, x ≤ 11/12; 4x+1 ≥ -14/3, 4x ≥ -14/3-1, 4x ≥ -17/3, x ≥ -17/12; x ≤ 11/12 or x ≥ -17/12, [-17/12,11/12], -17/12 ≤ x ≤ 11/12

-3 ≤ 2x+5 < 19; -3-5 ≤ 2x+5-5 < 19-5, -8 ≤ 2x < 14, -8/2 ≤ 2x/2 < 14/2, -4 ≤ x < 7, [-4,7)

x^{2}-x ≥ 12; x^{2}-x-12 ≥ 0, (x-4)(x+3) ≥ 0, x-4 = 0, x = 4, x+3 = 0, x = -3; if x = 5 then (5-4)(5+3) = 1⋅8 = 8 > 0; if x = 0 then (0-4)(0+3) = -4⋅3 = -12 < 0; if x = -4 then (-4-4)(-4+3) = -8⋅-12 = 96 > 0; x ≤ -3 or x ≥ 4, (-∞,-3]∪[4,+∞)

x^{2}+9 > 6x; x^{2}-6x+9 > 0, (x-3)(x-3) > 0, (x-3)^{2} > 0, x-3 = 0, x = 3; if x = 4 then (4-3)^{2} = 1^{2} = 1 > 0; if x = 2 then (2-3)^{2} = -1^{2} = 1 > 0; x < 3 or x > 3, (-∞,3)∪(3,+∞)

2x^{2} > x+6; 2x^{2}-x-6 > 0, 2x^{2}-4x+3x-6 > 0, 2x(x-2)+3(x-2) > 0, (x-2)(2x+3) > 0, x-2 = 0, x = 2, 2x+3 = 0, 2x = -3, x = -3/2; if x = -2 then (-2-2)(2⋅-2+3) = -4(-4+3) = -4⋅-1 = 4 > 0; if x = 0 then (0-2)(2⋅0+3) = -2(0+3) = -2⋅3 = -6 < 0; if x = 3 then (3-2)(2⋅3+3) = 1(6+3) = 1⋅9 = 9 >0; x < -3/2 or x > 2, (-∞,-3/2)∪(2,+∞); if 2x^{2} < x+6 then -3/2 < x < 2, (-3/2,2)