Solving Inequalities - SAT Subject Test in Math I

Card 0 of 6

Question

Give the solution set of the inequality

$\frac{2x-5}{x-7} > 0$

Answer

Two numbers of like sign have a positive quotient.

Therefore, $\frac{2x-5}{x-7} > 0$ has as its solution set the set of points at which and are both positive or both negative.

To find this set of points, we identify the zeroes of both expressions.

$x = 2\frac{1}{2}$

Since $\frac{2x-5}{x-7}$ is nonzero we have to exclude $x = 2\frac{1}{2}$ ; is excluded anyway since it would bring about a denominator of zero. We choose one test point on each of the three intervals $\left (-\infty, 2\frac{1}{2} \right ) , \left ( 2\frac{1}{2}, 7 \right ) , \left ( 7, \infty\right )$ and determine where the inequality is correct.

$\left (-\infty, 2\frac{1}{2} \right )$

Choose :

$\frac{2 \cdot 0-5}{0-7} = \frac{-5}{-7} = \frac{5}{7} > 0$ - True.

$\left ( 2\frac{1}{2}, 7 \right )$

Choose :

$\frac{2 \cdot 3-5}{3-7} = \frac{6}{-4} =- \frac{3}{2} > 0$ - False.

$\left ( 7, \infty\right )$

Choose :

$\frac{2 \cdot 8-5}{8-7} = \frac{11}{1} =11> 0$ - True.

The solution set is $\left (-\infty, 2\frac{1}{2} \right ) \cup \left ( 7, \infty\right )$

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Question

Solve for x.

$2x-7\geq 1$

Answer

Solving inequalities is very similar to solving an equation. We must start by isolating x by moving the terms farthest from it to the other side of the inequality. In this case, add 7 to each side.

$2x-7+7\geq 1+7$

$2x\geq 8$

Now, divide both sides by 2.

$x\geq 4$

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Question

Solve for x.

$3x+2\geq -7$

Answer

Solving inequalities is very similar to solving an equation. We must start by isolating x by moving the terms farthest from it to the other side of the inequality. In this case, subtract 2from each side.

$3x+2-2\geq -7-2$

$3x\geq -9$

Now, divide both sides by 2.

$x\geq -3$

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Question

Solve the following inequality:

Answer

To solve for an inequality, you solve like you would for a single variable expression and get by itself.

First, subtract from both sides to get,

.

Then divide both sides by and your final answer will be,

$x< \frac{2}{3}$ .

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Question

Solve the inequality:

Answer

Simplify the left side.

The inequality becomes:

Divide by two on both sides.

$\frac{2}{2}x<\frac{10}{2}$

The answer is:

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Question

Solve the inequality: $2x+6\leq 9x-3$

Answer

Subtract on both sides.

$2x+6-2x\leq 9x-3-2x$

$6\leq 7x-3$

Add 3 on both sides.

$6+3\leq 7x-3+3$

$9\leq 7x$

Divide by 7 on both sides.

$\frac{9}{7}\leq \frac{7x}{7}$

$\frac{9}{7}\leq x$

The answer is: $x\geq\frac{9}{7}$

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