Digital SAT Math Practice Test 8

An equation has infinitely many solutions when the left side is mathematically identical to the right side for all values of \(x\).

Step 1: Simplify the left side

\[ 4(x - 3) + 2x = 4x - 12 + 2x \]

\[ 6x - 12 = cx - 12 \]

Step 2: Compare coefficients

For the two sides to be identical, the coefficient of \(x\) on the left (6) must match the coefficient of \(x\) on the right (\(c\)).

Therefore, \(c = 6\).

Step 1: Determine the y-intercept (\(b\))

If a line passes through the origin \((0, 0)\), its y-intercept is exactly 0. Thus, \(b = 0\).

Step 2: Determine the slope (\(m\))

Using the origin and \((4, -2)\):

\[ m = \dfrac{y_2 - y_1}{x_2 - x_1} = \dfrac{-2 - 0}{4 - 0} = -0.5 \]

Step 3: Find the sum

\[ m + b = -0.5 + 0 = -0.5 \]

To solve the compound inequality, isolate \(x\) in the middle term:

Step 1: Add 5 to all parts

\[ -3 + 5 < 2x < 7 + 5 \Rightarrow 2 < 2x < 12 \]

Step 2: Divide all parts by 2

\[ 1 < x < 6 \]

The integer values for \(x\) that satisfy this are 2, 3, 4, and 5. The greatest possible integer value is 5.

The y-intercept occurs where \(x = 0\). Simply evaluate the function at 0:

\[ f(0) = \dfrac{0^2 - 9}{0 - 3} = \dfrac{-9}{-3} = 3 \]

(Alternatively, simplifying the function shows \(f(x) = x + 3\) for \(x \neq 3\), confirming the y-intercept is 3).

Apply the exponent rule \(a^{m+n} = a^m \cdot a^n\):

\[ (2^x \cdot 2^3) - 2^x = 112 \]

\[ 8(2^x) - 1(2^x) = 112 \]

Factor out the common term \(2^x\):

\[ 2^x(8 - 1) = 112 \Rightarrow 7(2^x) = 112 \]

Divide by 7: \[ 2^x = 16 \]

Since \(16 = 2^4\), the value of \(x\) is 4.

Step 1: Find the x-coordinate of the vertex

Using \(x = -\dfrac{b}{2a}\) with \(a = -2\) and \(b = 8\):

\[ x = -\dfrac{8}{2(-2)} = 2 \]

Step 2: Find the y-coordinate

Substitute \(x = 2\) into the original equation:

\[ y = -2(2)^2 + 8(2) - 5 = -8 + 16 - 5 = 3 \]

The y-coordinate of the vertex is 3.

Calculate the number of doubling periods: \(12 / 4 = 3\).

Applying the growth formula \(P = P_0 \times 2^n\):

\[ P = 150 \times 2^3 = 150 \times 8 = 1200 \]

The population after 12 hours will be 1200.

Set up a proportion: \[ \dfrac{2 \text{ sugar}}{3 \text{ flour}} = \dfrac{x \text{ sugar}}{15 \text{ flour}} \]

Cross-multiply: \[ 3x = 30 \Rightarrow x = 10 \]

The baker needs 10 cups of sugar.

Original total sum: \(5 \times 10 = 50\).

New total sum: \(6 \times 12 = 72\).

Value of the 6th number: \(72 - 50 = 22\).

A cube has 6 faces. Area of one face: \(54 / 6 = 9\).

Side length \(s = \sqrt{9} = 3\).

Volume \(V = s^3 = 3^3 = 27 \text{ cubic inches}\).

In a right triangle, acute angles \(A\) and \(B\) are complementary. This leads to the identity \(\sin(A) = \cos(90 - A)\), which is \(\cos(B)\).

Therefore, if \(\sin(A) = 0.6\), then \(\cos(B) = 0.6\).

Total circumference: \(C = 2\pi(12) = 24\pi\).

Fraction of the circle: \(60/360 = 1/6\).

Arc Length: \((1/6) \times 24\pi = 4\pi\).