Algebra Examples

$10^{2 x + 7} \geq 1000^{x}$

Step 1

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

$10^{2 x + 7} \geq 10^{3 x}$

Step 2

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

$2 x + 7 \geq 3 x$

Step 3

Solve for $x$ .

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Step 3.1

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

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Step 3.1.1

Subtract $3 x$ from both sides of the inequality.

$2 x + 7 - 3 x \geq 0$

Step 3.1.2

Subtract $3 x$ from $2 x$ .

$- x + 7 \geq 0$

Step 3.2

Subtract $7$ from both sides of the inequality.

$- x \geq - 7$

Step 3.3

Divide each term in $- x \geq - 7$ by $- 1$ and simplify.

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Step 3.3.1

Divide each term in $- x \geq - 7$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \leq \frac{- 7}{- 1}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} \leq \frac{- 7}{- 1}$

Step 3.3.2.2

Divide $x$ by $1$ .

$x \leq \frac{- 7}{- 1}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Divide $- 7$ by $- 1$ .

$x \leq 7$

Step 4

The result can be shown in multiple forms.

Inequality Form:

$x \leq 7$

Interval Notation:

$(- \infty, 7]$

Step 5