1.
Solving Equations: X+B = C
Solve for x:x + 3 = -1
Correct Answer
C. X = -4
Explanation
To solve the equation x + 3 = -1, we need to isolate the variable x on one side of the equation. To do this, we can subtract 3 from both sides of the equation. This gives us x = -4. Therefore, the correct answer is x = -4.
2.
Solving Equations: X+B = C
Solve for x:x + 7 = 9
Correct Answer
B. X = 2
Explanation
To solve the equation x + 7 = 9, we need to isolate the variable x on one side of the equation. To do this, we can subtract 7 from both sides of the equation. This will give us x = 2, as the value of x that satisfies the equation.
3.
Solving Equations: AX = C
Solve for x:8x = 4
Correct Answer
A. X = 1/2
Explanation
To solve the equation 8x = 4, divide both sides of the equation by 8. This will isolate the variable x on one side of the equation. Simplifying, we get x = 1/2.
4.
Solving Equations: AX = C
Solve for x:-6x = -2
Correct Answer
D. X = 1/3
Explanation
The given equation is -6x = -2. To solve for x, we need to isolate x on one side of the equation. By dividing both sides of the equation by -6, we get x = 1/3. Therefore, the correct answer is x = 1/3.
5.
Solving Equations: AX+B = C
Solve for x:19.74x - 1 = -357.8
Correct Answer
C. X = -17840/987
Explanation
The given equation is 19.74x - 1 = -357.8. To solve for x, we need to isolate the variable x on one side of the equation. We can do this by adding 1 to both sides of the equation, which gives us 19.74x = -356.8. Then, we divide both sides of the equation by 19.74 to solve for x. Simplifying the division, we get x = -17840/987.
6.
Solving Equations: AX+B = CX+D
Solve for x:-16.87x - 8 = 17.99x + 618
Correct Answer
B. X = -31300/1743
Explanation
The given equation is a linear equation of the form AX+B = CX+D. To solve for x, we need to isolate the variable on one side of the equation. By subtracting 17.99x from both sides and adding 8 to both sides, we get -16.87x - 17.99x = 618 + 8. Combining like terms, we have -34.86x = 626. Dividing both sides by -34.86, we get x = 626/-34.86. Simplifying the fraction gives x = -31300/1743.
7.
Solving Equations: AX+B = CX+D
Solve for x:-16.84x - 57.16 = 10.1x - 264.6
Correct Answer
A. X = 10372/1347
Explanation
The equation -16.84x - 57.16 = 10.1x - 264.6 is solved by first combining like terms on both sides of the equation. This gives us -16.84x - 10.1x = -264.6 + 57.16. Simplifying further, we get -26.94x = -207.44. To solve for x, we divide both sides of the equation by -26.94, which gives us x = -207.44/-26.94. Simplifying this expression, we get x = 10372/1347.
8.
Interval Notation 1
Write in interval notation:
<= is < and >= >
infinity = ∞3 < x <= 9
Correct Answer
B. ( 3, 9 ]
Explanation
The correct answer is ( 3, 9 ] because the interval notation represents all real numbers greater than 3 and less than or equal to 9. The square bracket on the right side indicates that 9 is included in the interval.
9.
Interval Notation 1
Write in interval notation:
<= is < and >= >
infinity = ∞x <= 3 or x > 8
Correct Answer
C. ( - infinity, 3 ] U ( 8, infinity )
Explanation
The given answer, ( - infinity, 3 ] U ( 8, infinity ), represents the interval notation for the given conditions. It includes all real numbers less than or equal to 3, as well as all real numbers greater than 8. The square bracket on the left side of 3 indicates that 3 is included in the interval, while the parentheses on the right side of 8 indicate that 8 is not included in the interval. Therefore, the answer accurately represents the given conditions in interval notation.
10.
Interval Notation 2
Write in set-builder notation:
<= is < and >= >
infinity = ∞( - infinity, -3 ] U ( 3, infinity )
Correct Answer
D. { x | x 3 }
Explanation
The correct answer is { x | x > 3 }. This set-builder notation represents all values of x that are greater than 3.
11.
Interval Notation 2
Write in set-builder notation:
<= is < and >= >
infinity = ∞[ 3, 7 ]
Correct Answer
A. { x | 3
12.
Solving Inequalities: AX+B < C
Solve for x:8x - 2 > -8
Correct Answer
A. X > -3/4
Explanation
To solve the inequality 8x - 2 > -8, we can start by adding 2 to both sides of the inequality to isolate the term with x. This gives us 8x > -6. Then, we divide both sides of the inequality by 8 to solve for x. Since dividing by a positive number does not change the direction of the inequality, we have x > -3/4. This means that any value of x greater than -3/4 will satisfy the inequality.
13.
Solving Inequalities: AX+B < C
Solve for x:2x - 2 > -3
Correct Answer
C. X > -1/2
Explanation
To solve the inequality 2x - 2 > -3, we can start by adding 2 to both sides of the inequality to isolate the variable term. This gives us 2x > -1. Then, we divide both sides of the inequality by 2 to solve for x. Since we are dividing by a positive number, the direction of the inequality remains the same. Therefore, x > -1/2 is the correct solution.
14.
Solving Inequalities: AX+B , CX+D
Solve for x:-x + 10 < -8x - 4
Correct Answer
D. X < -2
Explanation
To solve the inequality, we need to isolate the variable x. First, we can simplify the equation by adding 8x to both sides and subtracting 10 from both sides. This gives us 7x > -6. To isolate x, we divide both sides of the inequality by 7. However, since we are dividing by a negative number, the inequality sign must be flipped. Therefore, x < -2 is the correct answer.
15.
Solving Inequalities: AX+B , CX+D
Solve for x:-5x - 6 < x + 3
Correct Answer
A. X > -3/2
Explanation
To solve the inequality, we need to isolate the variable x. We can do this by subtracting x from both sides of the inequality: -5x - 6 - x < x + 3 - x. Simplifying this gives us -6x - 6 < 3. Next, we can add 6 to both sides: -6x - 6 + 6 < 3 + 6. Simplifying further gives us -6x < 9. Finally, we can divide both sides by -6, remembering to reverse the inequality sign because we are dividing by a negative number: -6x/-6 > 9/-6. Simplifying this gives us x > -3/2. Therefore, the correct answer is x > -3/2.
16.
Finding the Slope of a Line Given Two Points
Find the slope of the line passing through:( 0, -9 ) and ( 7, -10 )
Correct Answer
A. M = -1/7
Explanation
The slope of a line can be found using the formula (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are the coordinates of two points on the line. In this case, the coordinates are (0, -9) and (7, -10). Plugging these values into the formula, we get (-10 - (-9)) / (7 - 0) = -1/7. Therefore, the slope of the line passing through these two points is -1/7.
17.
Finding the Slope of a Line Given Two Points
Find the slope of the line passing through:( 7, 9 ) and ( -8, -4 )
Correct Answer
C. M = -13/-15 = 13/15
Explanation
The slope of a line can be found using the formula (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are the coordinates of two points on the line. In this case, the coordinates given are (7, 9) and (-8, -4). Plugging these values into the formula, we get (9 - (-4)) / (7 - (-8)) = 13 / 15. Therefore, the slope of the line passing through these two points is 13/15.
18.
Equation of a Line Given a Point & the Slope
Find the equation of the line:
with the slope
m = 1
and passing through the point( -1, -8 )
Correct Answer
A. -11x + 11y = -77
Explanation
The equation of a line can be determined using the point-slope form, which is y - y₁ = m(x - x₁), where (x₁, y₁) is the given point and m is the slope. In this case, the given point is (-1, -8) and the slope is 1. Plugging these values into the point-slope form, we get y - (-8) = 1(x - (-1)), which simplifies to y + 8 = x + 1. Rearranging the equation, we get x - y = -9, which can be rewritten as -11x + 11y = -77. Therefore, the correct answer is -11x + 11y = -77.
19.
Equation of a Line Given a Point & the Slope
Find the equation of the line:
with the slope
m = 6
and passing through the point( 3, 3 )
Correct Answer
D. Y = 6x - 15
Explanation
The equation of a line can be determined using the point-slope form, which is y - y1 = m(x - x1). In this case, the given slope is 6 and the point it passes through is (3, 3). Plugging these values into the point-slope form, we get y - 3 = 6(x - 3). Simplifying this equation gives us y = 6x - 15, which matches the given answer.
20.
Parallel Lines
Find the equation of the line that is parallel to the line:
9x - 3y = 7
and passes through the point:( -3, -1 )
Correct Answer
D. -3x + y = 8
Explanation
The equation -3x + y = 8 is the correct answer because it represents a line that is parallel to the given line 9x - 3y = 7. The reason for this is that the slope of both lines is the same. The given line can be rewritten in slope-intercept form as y = (9/3)x - 7/3, which has a slope of 9/3. The equation -3x + y = 8 can also be rewritten in slope-intercept form as y = 3x + 8, which also has a slope of 9/3. Therefore, both lines have the same slope and are parallel to each other. Additionally, the equation -3x + y = 8 passes through the point (-3, -1) as required.
21.
Parallel Lines
Find the equation of the line that is parallel to the line:
-7x + 4y = 9
and passes through the point:( 8, 9 )
Correct Answer
A. -7x + 4y = -20
Explanation
The equation -7x + 4y = -20 is the correct answer. Two lines are parallel if they have the same slope. The given line has a slope of -7/4, so any line with the same slope will be parallel to it. The equation -7x + 4y = -20 has the same slope as -7x + 4y = 9, and it also passes through the point (8, 9), making it the equation of the line that is parallel to the given line.
22.
Perpendicular Lines
Find the equation of the line that is perpendicular to the line:
-6x - 2y = -3
and passes through the point:
( -4, 7 )Find the slope by solving for y.
Correct Answer
D. -x + 3y = 25
Explanation
The given equation, -x + 3y = 25, is the equation of the line that is perpendicular to the line -6x - 2y = -3 and passes through the point (-4, 7). This is determined by finding the slope of the given line and taking its negative reciprocal. Then, using the point-slope form of a line, the equation is found by plugging in the coordinates of the point and the slope.
23.
Perpendicular Lines
Find the equation of the line that is perpendicular to the line:
-8x - 8y = -2
and passes through the point:
( -10, -2 )Find the slope by solving for y.
Correct Answer
C. -x + y = 8
Explanation
To find the equation of a line that is perpendicular to another line, we need to find the negative reciprocal of the slope of the given line. The given line has a slope of -8/8, which simplifies to -1. The negative reciprocal of -1 is 1. Therefore, the perpendicular line will have a slope of 1. We also know that the perpendicular line passes through the point (-10, -2). Using the point-slope formula, we can write the equation of the line as y - (-2) = 1(x - (-10)), which simplifies to y + 2 = x + 10. Rearranging the terms, we get the equation -x + y = 8.
24.
Function Notation 1
x^{2} = x^2
Given the function:
f ( x ) = -9x^2 + 2x + 7
findf ( 4 )
Correct Answer
C. F ( 4 ) = -129
Explanation
The given function is f(x) = -9x^2 + 2x + 7. To find f(4), we substitute 4 in place of x in the function. So, f(4) = -9(4)^2 + 2(4) + 7 = -9(16) + 8 + 7 = -144 + 8 + 7 = -129.
25.
Function Notation 1
x^{2} = x^2
Given the function:
f ( x ) = -3x^2 + 4x + 3
findf ( 9 )
Correct Answer
B. F ( 9 ) = -204
Explanation
The given function is f(x) = -3x^2 + 4x + 3. To find f(9), we substitute 9 in place of x in the function. So, f(9) = -3(9)^2 + 4(9) + 3. Simplifying this expression gives us -3(81) + 36 + 3 = -243 + 36 + 3 = -204. Therefore, the correct answer is f(9) = -204.
26.
Polynomials: Exponent Rules 1
Simplify:
notation: x^2 = x^{2}( x^7) ( w) ( x^6) ( y^3) ( w^6) ( x^5)
Correct Answer
A. ( w^7) ( x^18) ( y^3)
27.
Polynomials: Exponent Rules 1
Simplify:
notation: x^2 = x^{2}( x) ( w^9) ( x^8) ( y^5) ( w^6) ( x)
Correct Answer
B. ( w^15) ( x^10) ( y^5)
28.
Polynomials: Exponent Rules 2
Simplify:
notation: x^2 = x^{2}
62 ( x^9) ( y^7)89 ( x^3) ( y^3)
Correct Answer
B. 62 (x^6) (y^4)
89
29.
Polynomials: Exponent Rules 2
Simplify:
notation: x^2 = x^{2}
80 ( x^4) ( w^5) ( x^2)47 ( x^2) ( w^4) ( w^8)
Correct Answer
A. 80 (x^4)
47 (w^7)
30.
Polynomials: Adding & Subtracting
notation: a^2b = a^{2}b
Perform the operation:( -2x^2 + 1 ) - ( 6x^3 - 9x - 4 )
Correct Answer
A. -6x^3 - 2x^2 + 9x + 5
Explanation
The given expression is (-2x^2 + 1) - (6x^3 - 9x - 4). To subtract the polynomials, we need to change the signs of the second polynomial and then combine like terms. After performing the subtraction, we get -6x^3 - 2x^2 + 9x + 5 as the result.
31.
Polynomials: Adding & Subtracting
notation: a^2b = a^{2}b
Perform the operation:( 6x^2 + 60x - 9 ) - ( -4x^2 + 5x - 1 )
Correct Answer
D. 10x^2 + 55x - 8
Explanation
The given expression is a subtraction of two polynomials. To subtract polynomials, we need to combine like terms. In this case, we have:
(6x^2 + 60x - 9) - (-4x^2 + 5x - 1)
When we remove the parentheses and combine like terms, we get:
6x^2 + 60x - 9 + 4x^2 - 5x + 1
Simplifying further, we combine the x^2 terms, the x terms, and the constant terms separately:
(6x^2 + 4x^2) + (60x - 5x) + (-9 + 1)
This gives us:
10x^2 + 55x - 8
Therefore, the correct answer is 10x^2 + 55x - 8.
32.
Multiplying Polynomials: FOIL
Multiply:
x^{2} = x^2( 5x - 3 )( 4x + 5 )
Correct Answer
B. 20x^2 + 13x – 15
Explanation
The given expression involves multiplying two binomials using the FOIL method. FOIL stands for First, Outer, Inner, Last, which means multiplying the first terms, then the outer terms, then the inner terms, and finally the last terms. In this case, the first terms are x^2 and 5x, the outer terms are x^2 and 5, the inner terms are -3 and 4x, and the last terms are -3 and 5. Simplifying each of these multiplications and combining like terms, we get the answer: 20x^2 + 13x - 15.
33.
Multiplying Polynomials: FOIL
Multiply:
x^{2} = x^2( 2x + 7 )( 2x + 2 )
Correct Answer
A. 4x^2 + 18x + 14
34.
Factoring: X^2 Trinomials
x^{2} = x^2
Factor:x^2 + 8x – 48
Correct Answer
B. (x + 12) (x - 4)
Explanation
The given expression is a quadratic trinomial in the form of ax^2 + bx + c. To factor it, we need to find two numbers that multiply to give c (in this case, -48) and add up to give b (in this case, 8). The numbers that satisfy this condition are 12 and -4. Therefore, the expression can be factored as (x + 12) (x - 4).
35.
Factoring: X^2 Trinomials
x^{2} = x^2
Factor:x^2 - 7x + 6
Correct Answer
A. (x - 1) (x - 6)
Explanation
The given expression is a quadratic trinomial in the form of ax^2 + bx + c. To factor it, we need to find two binomials that, when multiplied together, will give us the original trinomial. In this case, we need to find two binomials whose product will be x^2 - 7x + 6. By trial and error, we find that (x - 1) and (x - 6) are the correct binomials. When we multiply them together, we get x^2 - 7x + 6, which matches the original trinomial. Therefore, the correct answer is (x - 1) (x - 6).
36.
Factoring: aX^2 Trinomials
x^{2} = x^2
Factor:x^2 - 5x + 4
Correct Answer
B. (x - 4)(x - 1)
Explanation
The given expression is a trinomial in the form of ax^2 + bx + c. In order to factor it, we need to find two binomials that, when multiplied together, result in the given trinomial. In this case, the factors are (x - 4) and (x - 1), which when multiplied together, give us x^2 - 5x + 4. Therefore, the correct answer is (x - 4)(x - 1).
37.
Factoring: Difference of Two Squares
x^{2} = x^2
Factor:x^2 – 4
Correct Answer
C. (x + 2)(x - 2)
Explanation
The correct answer is (x + 2)(x - 2) because the given expression, x^2 - 4, can be factored as the difference of two squares. The square root of 4 is 2, so we can rewrite the expression as (x^2 - 2^2). Using the formula for the difference of two squares, we can factor it as (x + 2)(x - 2).
38.
Factoring: Difference of Two Squares
x^{2} = x^2
Factor:4x^2 – 1
Correct Answer
D. (2x + 1)(2x - 1)
Explanation
The given expression, 4x^2 - 1, can be factored using the difference of two squares formula. The formula states that a^2 - b^2 can be factored as (a + b)(a - b). In this case, a = 2x and b = 1. Plugging these values into the formula, we get (2x + 1)(2x - 1).
39.
Factoring: Factor by Grouping
x^{2} = x^2
Factor:7ab + 2ax - 28b - 8x
Correct Answer
A. ( a - 4 )( 7b + 2x )
40.
Factoring: Factor by Grouping
x^{2} = x^2
Factor:14xy - 18by - 21ax + 27ab
Correct Answer
B. ( 2y - 3a )( 7x - 9b )
Explanation
The given expression can be factored by grouping. We can group the terms as (14xy - 18by) and (-21ax + 27ab).
In the first group, we can factor out 2y, which gives us 2y(7x - 9b).
In the second group, we can factor out -3a, which gives us -3a(7x - 9b).
Now we have (2y - 3a)(7x - 9b), which is the factored form of the expression.
41.
Polynomials: Exponent Rules 5
Simplify:
notation: x^2*w^3 = x^{2}w^{3}( w^7 * y^5 * x )^3
Correct Answer
C. ( w^21) ( y^15) ( x^3)
42.
Polynomials: Exponent Rules 5
Simplify:
notation: x^2*w^3 = x^{2}w^{3}( w * y^7 * x )^3
Correct Answer
D. ( w^3) ( y^21) ( x^3)
Explanation
The given expression is a product of several terms, each containing variables raised to different exponents. To simplify the expression, we can combine the terms that have the same variable and add their exponents. In this case, we have variables x, y, and w. For x, we have x^2 in the first term and x^3 in the last term. When we multiply these together, we get x^(2+3) = x^5. Similarly, for y, we have y^7 in the first term and y^21 in the last term. Multiplying these together gives us y^(7+21) = y^28. Finally, for w, we have w^3 in both the first and last terms, so we simply keep w^3. Therefore, the simplified expression is (w^3)(y^21)(x^5).
43.
Polynomials: Exponent Rules 5 (with negative exponents)
Simplify:
notation: x^2*w^3 = x^{2}w^{3}( 4 x^-9 * y^3 )^-3
Correct Answer
A. (x^27)
64 (y^9)
44.
Polynomials: Exponent Rules 5 (with negative exponents)
Simplify:
notation: x^2*w^3 = x^{2}w^{3}( x^-10 * y^10 )^4
Correct Answer
B. (y^40)
(x^40)
Explanation
The given expression involves simplifying a polynomial with negative exponents. To simplify, we can apply the exponent rules. When we raise a power to another power, we multiply the exponents. In this case, we have (x^-10 * y^10)^4. When we raise a negative exponent to an even power, it becomes positive. So, (x^-10)^4 becomes x^40. Similarly, (y^10)^4 becomes y^40. Therefore, the simplified expression is (x^40)(y^40).
45.
Solving Quadratics: By Factoring
x^{2} = x^2
Solve for x:16x^2 + 24x + 5 = 0
Correct Answer
C. X = -1/4 or x = -5/4
46.
Solving Quadratics: By Factoring
x^{2} = x^2
Solve for x:10x^2 - 9x - 9 = 0
Correct Answer
D. X = 3/2 or x = -3/5
47.
Solving Quadratics: By Factoring 2
x^{2} = x^2
Solve for x:3x^2 - 6x + 3 = 0
Correct Answer
A. X = 1 or x = 1
48.
Solving Quadratics: By Factoring 2
x^{2} = x^2
Solve for x:4x^2 - 8x + 3 = 0
Correct Answer
B. X = 3/2 or x = ½
Explanation
The given quadratic equation can be factored as (2x - 1)(2x - 3) = 0. This equation can be solved by setting each factor equal to zero and solving for x. Thus, we have 2x - 1 = 0, which gives x = 1/2, and 2x - 3 = 0, which gives x = 3/2. Therefore, the correct answer is x = 3/2 or x = 1/2.
49.
Completing the Square 1
Complete the square to get this thing in the form:
f(x) = (x-h)^{2} +kf (x) = x^2 + 18x + 82
Correct Answer
C. F (x) = ( x + 9 )^2 + 1
Explanation
The given quadratic function is in the form f(x) = (x-h)^2 + k, which is the vertex form of a quadratic equation. By comparing the given equation with the vertex form, we can determine that the vertex of the parabola is (-9, 1). Additionally, since the coefficient of (x-h)^2 is positive, the parabola opens upwards. Therefore, the correct answer is f(x) = (x + 9)^2 + 1.
50.
Completing the Square 1
Complete the square to get this thing in the form:
f(x) = (x-h)^{2} +kf (x) = x^2 - 12x + 45
Correct Answer
D. F (x) = ( x - 6 )^2 + 9
Explanation
The given quadratic equation is in the form f(x) = (x-h)2 + k, which is the standard form of a quadratic equation after completing the square. By comparing the given equation f(x) = x^2 - 12x + 45 with the standard form, we can see that h = 6 and k = 9. Therefore, the equation can be rewritten as f(x) = (x - 6)^2 + 9.