Algebra Examples

$10^{4 x + 1} \geq 100^{x - 2}$

Step 1

Create equivalent expressions in the equation that all have equal bases.

$10^{4 x + 1} \geq 10^{2 (x - 2)}$

Step 2

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

$4 x + 1 \geq 2 (x - 2)$

Step 3

Solve for $x$ .

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Step 3.1

Simplify $2 (x - 2)$ .

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Step 3.1.1

Rewrite.

$4 x + 1 \geq 0 + 0 + 2 (x - 2)$

Step 3.1.2

Simplify by adding zeros.

$4 x + 1 \geq 2 (x - 2)$

Step 3.1.3

Apply the distributive property.

$4 x + 1 \geq 2 x + 2 \cdot - 2$

Step 3.1.4

Multiply $2$ by $- 2$ .

$4 x + 1 \geq 2 x - 4$

Step 3.2

Move all terms containing $x$ to the left side of the inequality.

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Step 3.2.1

Subtract $2 x$ from both sides of the inequality.

$4 x + 1 - 2 x \geq - 4$

Step 3.2.2

Subtract $2 x$ from $4 x$ .

$2 x + 1 \geq - 4$

Step 3.3

Move all terms not containing $x$ to the right side of the inequality.

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Step 3.3.1

Subtract $1$ from both sides of the inequality.

$2 x \geq - 4 - 1$

Step 3.3.2

Subtract $1$ from $- 4$ .

$2 x \geq - 5$

Step 3.4

Divide each term in $2 x \geq - 5$ by $2$ and simplify.

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Step 3.4.1

Divide each term in $2 x \geq - 5$ by $2$ .

$\frac{2 x}{2} \geq \frac{- 5}{2}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $2$ .

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Step 3.4.2.1.1

Cancel the common factor.

$\frac{2 x}{2} \geq \frac{- 5}{2}$

Step 3.4.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{- 5}{2}$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Move the negative in front of the fraction.

$x \geq - \frac{5}{2}$

Step 4

Use each root to create test intervals.

$x < - \frac{5}{2}$

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

Step 5

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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Step 5.1

Test a value on the interval $x < - \frac{5}{2}$ to see if it makes the inequality true.

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Step 5.1.1

Choose a value on the interval $x < - \frac{5}{2}$ and see if this value makes the original inequality true.

$x = - 5$

Step 5.1.2

Replace $x$ with $- 5$ in the original inequality.

$10^{4 (- 5) + 1} \geq 100^{(- 5) - 2}$

Step 5.1.3

The left side $1 E - 19$ is equal to the right side $1 E - 14$ , which means that the given statement is always true.

True

Step 5.2

Test a value on the interval $x > - \frac{5}{2}$ to see if it makes the inequality true.

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Step 5.2.1

Choose a value on the interval $x > - \frac{5}{2}$ and see if this value makes the original inequality true.

$x = 0$

Step 5.2.2

Replace $x$ with $0$ in the original inequality.

$10^{4 (0) + 1} \geq 100^{(0) - 2}$

Step 5.2.3

The left side $10$ is greater than the right side $0.0001$ , which means that the given statement is always true.

True

Step 5.3

Compare the intervals to determine which ones satisfy the original inequality.

$x < - \frac{5}{2}$ True

$x > - \frac{5}{2}$ True

$x < - \frac{5}{2}$ True

$x > - \frac{5}{2}$ True

Step 6

The solution consists of all of the true intervals.

$x \leq - \frac{5}{2}$ or $x \geq - \frac{5}{2}$

Step 7

Combine the intervals for any value of $x$ .

All real numbers

Step 8

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$

Step 9