Basic Single-Variable Algebra

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Math › Basic Single-Variable Algebra

Questions 1 - 10

Solve for :

$3(x + 4) \leq x - 6$

$x \leq -9$

$x \leq 9$

$x \geq -9$

$x \geq 9$

$x \leq 3$

Explanation

The first step is to distribute (multiply) through the parentheses:

$3(x + 4) \leq x - 6$

$3x + 12 \leq x - 6$

Then subtract from both sides of the inequality:

$2x + 12 \leq -6$

Next, subtract the 12:

$2x \leq -18$

Finally, divide by two:

$x \leq -9$

Solve the inequality:

$\small x^2+4< 9$

$\small -\sqrt5< x< \sqrt5$

$\small x< \sqrt5$

$\small x< -\sqrt5$

$\small -\sqrt5> x\ or\ x> \sqrt5$

Explanation

$\small \small x^2+4< 9$

$\small x^2< 5$

$\small x< \left | (\pm\sqrt5) \right |$

$\small -\sqrt5< x< \sqrt5$

Solve the inequality:

$\small x^2+4< 9$

$\small -\sqrt5< x< \sqrt5$

$\small x< \sqrt5$

$\small x< -\sqrt5$

$\small -\sqrt5> x\ or\ x> \sqrt5$

Explanation

$\small \small x^2+4< 9$

$\small x^2< 5$

$\small x< \left | (\pm\sqrt5) \right |$

$\small -\sqrt5< x< \sqrt5$

Simplify the expression.

$\small \frac{(x^2y^2)(x^3y^4z)}{x^4y^3z^4}$

$\small \frac{xy^3}{z^3}$

$\small \frac{x^2y^5}{z^3}$

$\small \frac{z^3}{x^2y^5}$

$\small \frac{z^3}{xy^3}$

Explanation

$\small \frac{(x^2y^2)(x^3y^4z)}{x^4y^3z^4}$

Because we are only multiplying terms in the numerator, we can disregard the parentheses.

$\small \frac{(x^2y^2)(x^3y^4z)}{x^4y^3z^4}=\frac{x^2y^2x^3y^4z}{x^4y^3z^4}$

To combine like terms in the numerator, we add their exponents.

$\small \frac{x^5y^6z}{x^4y^3z^4}$

To combine like terms between the numerator and denominator, subtract the denominator exponent from the numerator exponent.

$x^{5-4}y^{6-3}z^{1-4}=xy^3z^{-3}$

Remember that any negative exponents stay in the denominator.

$\frac{xy^3}{z^3}$

varies inversely as the square root of . If , then . Find if (nearest tenth, if applicable).

Explanation

The variation equation is $A = \frac{K}{\sqrt{x}}$ for some constant of variation .

Substitute the numbers from the first scenario to find :

$45= \frac{K}{\sqrt{15}}$

$K = 45 \sqrt{15}$

The equation is now $A = \frac{45 \sqrt{15}}{\sqrt{x}}$ .

If , then

$A = \frac{45 \sqrt{15}}{\sqrt{30}} \approx 31.8$

Simplify the following expression by combining like terms:

Explanation

Simplify the following expression by combining like terms:

Begin by looking for terms to combine. In this case, we only have 2 terms we can combine. Remember, we can only combine terms that have the same exponent and variable.

${\color{Green} 4x^6}-5x^3+{\color{Green} 2x^6}-5$

In this case, the green colors are the only ones which can be combined:

So our answer is:

Simplify the following expression by combining like terms:

Explanation

Simplify the following expression by combining like terms:

Begin by looking for terms to combine. In this case, we only have 2 terms we can combine. Remember, we can only combine terms that have the same exponent and variable.

${\color{Green} 4x^6}-5x^3+{\color{Green} 2x^6}-5$

In this case, the green colors are the only ones which can be combined:

So our answer is:

In April, the price of a t-shirt is $\$X$ . In May, the store increases the price by 50%, so that the new price is $\$Y$ . Then in June, the store decreases the price by 50%, so that the t-shirt price is now $\$Z$ . What is the ratio of to ?

$\frac{3}{4}$

$\frac{4}{3}$

$\frac{1}{2}$

Explanation

If the original price of the T-shirt is , increasing the price by 50% means that the new price is 150% of , or .

If the price is then decreased by 50%, the new price is 50% of

$.5\cdot (1.5X)=.75X$

The ratio of to is then:

$\frac{.5*1.5*X}{X}$

The 's in the numerator and denominator cancel, leaving $.5\cdot 1.5$ , or

$\frac{3}{4}$ .

The distance of a cyclist is directly proportional to the time he has traveled. Suppose he has traveled 12 miles in 1.5 hours. How far does he travel in a half hour?

Explanation

Write the equation for direct proportionality.

$y=kx \rightarrow D=kT$

Substitute the distance and time given to solve for the constant of proportionality, .

Divide by 1.5 on both sides.

$\frac{12}{1.5} = \frac{k(1.5)}{1.5}$

Write the equation.

Substitute half an hour for the time to determine the distance the biker has traveled.

The biker traveled four miles in a half hour.

The answer is:

Tom is painting a fence feet long. He starts at the West end of the fence and paints at a rate of feet per hour. After hours, Huck joins Tom and begins painting from the East end of the fence at a rate of feet per hour. After hours of the two boys painting at the same time, Tom leaves Huck to finish the job by himself.

If Huck completes painting the entire fence after Tom leaves, how many more hours will Huck work than Tom?

$6\ hours$

$10\ hours$

$4\ hours$

$3\ hours$

$5\ hours$

Explanation

Tom paints for a total of hours (2 on his own, 2 with Huck's help). Since he paints at a rate of feet per hour, use the formula

$distance = rate \times time$ (or )

to determine the total length of the fence Tom paints.

feet

Subtracting this from the total length of the fence feet gives the length of the fence Tom will NOT paint: feet. If Huck finishes the job, he will paint that feet of the fence. Using , we can determine how long this will take Huck to do: