Algebra Examples

$- 4 x - 3 \leq 2 x + 9$ and $- 3 x + 2 > - 2 x + 5$

Step 1

Simplify the first inequality.

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Step 1.1

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

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Step 1.1.1

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

$- 4 x - 3 - 2 x \leq 9$ and $- 3 x + 2 > - 2 x + 5$

Step 1.1.2

Subtract $2 x$ from $- 4 x$ .

$- 6 x - 3 \leq 9$ and $- 3 x + 2 > - 2 x + 5$

Step 1.2

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

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Step 1.2.1

Add $3$ to both sides of the inequality.

$- 6 x \leq 9 + 3$ and $- 3 x + 2 > - 2 x + 5$

Step 1.2.2

Add $9$ and $3$ .

$- 6 x \leq 12$ and $- 3 x + 2 > - 2 x + 5$

Step 1.3

Divide each term in $- 6 x \leq 12$ by $- 6$ and simplify.

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Step 1.3.1

Divide each term in $- 6 x \leq 12$ by $- 6$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 6 x}{- 6} \geq \frac{12}{- 6}$ and $- 3 x + 2 > - 2 x + 5$

Step 1.3.2

Simplify the left side.

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Step 1.3.2.1

Cancel the common factor of $- 6$ .

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Step 1.3.2.1.1

Cancel the common factor.

$\frac{- 6 x}{- 6} \geq \frac{12}{- 6}$ and $- 3 x + 2 > - 2 x + 5$

Step 1.3.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{12}{- 6}$ and $- 3 x + 2 > - 2 x + 5$

Step 1.3.3

Simplify the right side.

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Step 1.3.3.1

Divide $12$ by $- 6$ .

$x \geq - 2$ and $- 3 x + 2 > - 2 x + 5$

Step 2

Simplify the second inequality.

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Step 2.1

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

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Step 2.1.1

Add $2 x$ to both sides of the inequality.

$x \geq - 2$ and $- 3 x + 2 + 2 x > 5$

Step 2.1.2

Add $- 3 x$ and $2 x$ .

$x \geq - 2$ and $- x + 2 > 5$

Step 2.2

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

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Step 2.2.1

Subtract $2$ from both sides of the inequality.

$x \geq - 2$ and $- x > 5 - 2$

Step 2.2.2

Subtract $2$ from $5$ .

$x \geq - 2$ and $- x > 3$

Step 2.3

Divide each term in $- x > 3$ by $- 1$ and simplify.

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Step 2.3.1

Divide each term in $- x > 3$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$x \geq - 2$ and $\frac{- x}{- 1} < \frac{3}{- 1}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Dividing two negative values results in a positive value.

$x \geq - 2$ and $\frac{x}{1} < \frac{3}{- 1}$

Step 2.3.2.2

Divide $x$ by $1$ .

$x \geq - 2$ and $x < \frac{3}{- 1}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Divide $3$ by $- 1$ .

$x \geq - 2$ and $x < - 3$

Step 3

The intersection consists of the elements that are contained in both intervals.

No solution