Solving for the Variable - GED Math

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Question

Give the solution set of the inequality:

$7y + 10 \leq -3y + 24$

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Answer

$7y + 10 \leq -3y + 24$

$7y + 10 + 3y \leq -3y + 24 + 3y$

$10y + 10 \leq 24$

$10y + 10- 10 \leq 24 - 10$

$10y \leq 14$

$10y \div 10 \leq 14 \div 10$

$y \leq 1.4$

or, in interval form, $\left ( -\infty, 1.4 \right ]$

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Question

Give the solution set of the inequality:

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Answer

$4y\div 4 > 2\div 4$

$y > $\frac{1}{2}$$

In interval form, this is $\left ( $\frac{1}{2}$, \infty \right )$ .

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Question

Solve for :

$\small $\frac{n}{2}$+4=10$

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Answer

$\small $\frac{n}{2}$+4=10$

$\small 2($\frac{n}{2}$+4)=2(10)$

$\small n+8=20$

$\small n+8-8=20-8$

$\small n=12$

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Question

Solve for :

$\small $\frac{n}{2}$+4=3$

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Answer

$\small $\frac{n}{2}$+4=3$

$\small 2($\frac{n}{2}$+4)=2(3)$

$\small n+8=6$

$\small n+8-8=6-8$

$\small n=-2$

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Question

Which of the following is the solution set of the inequality $-13 \leq 8-3x \leq 20$ ?

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Answer

Solve using the properties of inequality, as follows:

$-13 \leq 8-3x \leq 20$

$-13- 8 \leq 8-3x - 8 \leq 20 - 8$

$-21 \leq -3x \leq 12$

$-21 \div (-3) \geq -3x \div (-3) \geq 12 \div (-3)$

Note that division by a negative number reverses the symbols.

$7 \geq x\geq -4$

In interval form, this is $\left [ -4, 7\right ]$ .

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Question

Two more than twice a number equals 9. What's the square of that number?

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Answer

Rewrite the algebraic expression in a mathematical formula.

Solve for x.

$x= $\frac{7}{2}$$

The square of this number is:

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Question

If , then what is the value of ?

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Answer

In order to solve for the value of you must isolate the variable. This is done by subtracting the constant in this equation, which is 12, from both sides of the equation.

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Question

If , what is the value of ?

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Answer

The first step in the process of solving for in this problem is to use the distributive property to distribute the to what is inside the parentheses.

The next step is to isolate the variable by using inverse operations. In this example, in order to get rid of the , you would add to both sides of the equation.

The next step is to divide both sides by the coefficient, (the number next to the variable), which in this case is .

$$\frac{5x}{5}$ = x$

$$\frac{75}{5}$ = 15$

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Question

If , then

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Answer

To solve this you must find the value of .

The first equation states that . This is a mult-step equation. The first step is to remove the constant, 6, from the equation; this is done by using the inverse operation, which means you would subtract the 6 from both sides of the equation.

Then divide both sides by the 7 in order to isolate the variable.

$\frac{7x}{7}$ $=\frac{21}{7}$$

Then plug the 3 into the second equation for the value of x.

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Question

Solve for .

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Answer

$5 \cdot y + 5 \cdot 4 +2 \cdot y +2 \cdot 2 = 7y$

Since the original statement forces this false statement to be true, the original statement is false regardless of the value of . There is no solution.

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Question

Solve the inequality for :

$6-$\frac{x}{3}$ > 8$

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Answer

$6-$\frac{x}{3}$ > 8$

$6-$\frac{x}{3}$- 6 > 8 - 6$

$-$\frac{x}{3}$ > 2$

$-$\frac{x}{3}$ \cdot (-3) < 2 \cdot (-3)$ - note the switch in the inequality symbol

That is, $\left ( -\infty ,-6 \right )$ .

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Question

$10 > 7-$\frac{x}{4}$ > 5$

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Answer

$10 > 7-$\frac{x}{4}$ > 5$

$10 - 7 > 7-$\frac{x}{4}$ - 7 > 5 - 7$

$3 > -$\frac{x}{4}$ > -2$

$3 \cdot (-4) < -$\frac{x}{4}$ \cdot (-4) < -2 \cdot (-4)$

Note the switch in the inequality symbol.

.

This can also be written as .

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Question

Give the solution set of the inequality:

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Answer

$-10y \div (-10) < 2 \div (-10)$

Note that the inequality symbol changes.

$y < -$\frac{2}{10}$$

$y < -$\frac{1}{5}$$

or, in interval notation, $\left ( -\infty, -$\frac{1}{5}$ \right )$ .

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Question

Solve for :

$\frac{1}{4}$ t = $\frac{5}{8}$

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Answer

$\frac{1}{4}$ t = $\frac{5}{8}$

Multiply both sides by 4 to isolate :

$\frac{4}{1}$\cdot $\frac{1}{4}$ t =\frac{4}{1}$\cdot $\frac{5}{8}$

$t =\frac{4\cdot 5}{1 \cdot 8}$ = $\frac{20}{8}$ = $\frac{20 \div 4}{8 \div 4}$ = $\frac{5}{2}$$

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Question

Solve for :

$t + $\frac{1}{6}$ = $\frac{4}{3}$$

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Answer

$t + $\frac{1}{6}$ = $\frac{4}{3}$$

$t + $\frac{1}{6}$ - $\frac{1}{6}$ = $\frac{4}{3}$ - $\frac{1}{6}$$

$t = $\frac{4}{3}$ - $\frac{1}{6}$ = $\frac{4 \cdot 2 }{3 \cdot 2 }$ - $\frac{1}{6}$ = $\frac{8}{6}$ - $\frac{1}{6}$ = $\frac{7}{6}$$

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Question

Consider the expression $\frac{7-3N}{17 - 2N}$ .

What value, if substituted in for , makes this expression undefined?

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Answer

$\frac{7-3N}{17 - 2N}$ is undefined if and only if its denominator is equal to 0.

$\frac{2N }{2}$= $\frac{17}{2}$

$N = 8 $\frac{1}{2}$$

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Question

Solve for :

$-$\frac{2}{7}$ y \geq -4$

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Answer

$-$\frac{2}{7}$ y \geq -4$

$-$\frac{2}{7}$ y \cdot\left ( -$\frac{7}$ {2} \right ) \leq -4 \cdot\left ( -$\frac{7}$ {2} \right )$

Note that the inequality symbol changes.

$y\leq $\frac{28}$ {2}$

$y\leq 14$

or, in interval notation, $\left ( -\infty , 14\right ]$ .

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Question

Solve for :

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Answer

In order to solve for , we will need the equation to be in terms of , and isolate the variable .

Solve by grouping the terms together. Subtract on both sides.

Divide by negative five on both sides.

$\frac{-5x }{-5}$= $\frac{-2y}{-5}$

The answer is: $x=\frac{2}{5}$y$

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Question

Solve for :

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Answer

Distribute the term on the right side of the equation.

Combine like terms.

Subtract on both sides.

Divide by negative on both sides.

$\frac{18}{-18}$ = $\frac{-18x}{-18}$

The answer is:

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Question

Which of the following makes this equation true:

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Answer

To answer the question, we will solve for y. So, we get

$\frac{12y}{12}$ = $\frac{36}{12}$

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