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Difficulty
Simplify the expression and report the coefficient of $x$:
Solution
Combine the like terms:
So the simplified expression is
The coefficient of $x$ is $11$.
Solve for $x$:
Solution
Add $8$ to both sides:
Divide both sides by $5$:
Compute the value of
Solution
Follow the order of operations:
and
Then multiply:
Finally add:
After distributing, what is the coefficient of $x$ in the expression below?
Solution
Distribute the $4$ to each term:
The coefficient of $x$ is $12$.
If
what is $f(2)$?
Solution
Substitute $2$ for $x$:
Solve for $x$:
Solution
Subtract $2$ from both sides:
Multiply both sides by $6$:
Simplify the expression and give the coefficient of $x$:
Solution
The $x$-terms have a common denominator:
So the coefficient of $x$ is $1$.
Solve for $x$:
Solution
Divide both sides by $3$:
Subtract $4$ from both sides:
Solve for $x$:
Solution
Multiply both sides by $9$:
If
what is $g(3)$?
Solution
Substitute $3$ for $t$:
Evaluate the exponent first:
Difficulty
Solve for $x$:
Solution
Expand the left-hand side:
Combine constants:
Move variable terms to one side:
Add $7$:
Divide by $5$:
Solve for $x$:
Solution
Multiply the entire equation by $4$ to clear the denominators:
Distribute:
Combine like terms:
Subtract $5$ and divide by $3$:
In the system
what is the value of $x$?
Solution
Substitute $y = x + 2$ into the second equation:
Combine like terms:
Subtract $2$:
Divide by $2$:
The equation
has two solutions. What is the smaller solution?
Solution
Factor the quadratic:
Use the zero-product property:
So the solutions are $x = 3$ and $x = 4$.
The smaller solution is $3$.
Two points lie on a line:
What is the slope of the line?
Solution
Use the slope formula:
Let
and
What is the value of $f(g(2))$?
Solution
First evaluate the inner function:
Now substitute that result into $f$:
For $x \ne 3$, evaluate
at $x = 5$.
Solution
Factor the numerator:
So, for $x \ne 3$,
Now substitute $x = 5$:
Solve for $x$:
Solution
Expand the left-hand side:
Combine constants:
Subtract $3x$ from both sides:
Add $10$:
Divide by $5$:
Difficulty
A climbing gym charges an upfront fee of \$18 plus \$9 per visit.
If the total bill is \$72, how many visits did the student make?
Solution
Let $v$ be the number of visits. Set up the equation:
Subtract $18$:
Divide by $9$:
A rectangle has perimeter $54$ cm.
Its length is $5$ cm more than its width.
What is the width of the rectangle?
Solution
Let $w$ be the width. Then the length is $w + 5$.
Use the perimeter formula:
Expand and combine like terms:
Subtract $10$:
Divide by $4$:
At a concert, 2 tickets and 3 drinks cost \$34.
Also, 1 ticket and 1 drink cost \$13.
What is the price of one ticket?
Solution
Let $t$ be the ticket price and $d$ be the drink price.
Set up the system:
Multiply the second equation by $2$:
Subtract this from the first equation:
Substitute back into $t + d = 13$:
A water tank starts with $120$ liters.
It gains $25$ liters per hour.
After how many hours will it contain $245$ liters?
Solution
Let $h$ be the number of hours. Write the equation:
Subtract $120$:
Divide by $25$:
A recipe uses $3$ cups of flour for $8$ servings.
If the same recipe is scaled to $16$ servings, how many cups of flour are needed?
Solution
Since $16$ is twice $8$, the flour amount also doubles.
So the recipe needs $6$ cups of flour.
Difficulty
What value of $k$ makes
a perfect square trinomial?
Solution
Compare the expression to the square of a binomial:
Matching coefficients shows that
The line
has slope $-2$.
What is the slope of a line perpendicular to it?
Solution
Rewrite the line in slope-intercept form:
So the slope is $-2$.
A perpendicular line has slope equal to the negative reciprocal:
In decimal form, that is $0.5$.
The quadratic
has roots $4$ and $-3$.
What is the value of $c$?
Solution
Write the quadratic using its roots:
Expand:
The constant term is
Solve for $x$, making sure to respect any restrictions:
Solution
First note the restriction:
Factor the numerator:
So for $x \ne 5$,
Now solve:
