Algebra Examples

$- 2 a + 3 \geq 6 a - 1 > 3 a - 10$

Step 1

Solve $- 2 a + 3 \geq 6 a - 1$ for $a$ .

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Step 1.1

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

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Step 1.1.1

Subtract $6 a$ from both sides of the inequality.

$- 2 a + 3 - 6 a \geq - 1$

Step 1.1.2

Subtract $6 a$ from $- 2 a$ .

$- 8 a + 3 \geq - 1$

Step 1.2

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

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Step 1.2.1

Subtract $3$ from both sides of the inequality.

$- 8 a \geq - 1 - 3$

Step 1.2.2

Subtract $3$ from $- 1$ .

$- 8 a \geq - 4$

Step 1.3

Divide each term in $- 8 a \geq - 4$ by $- 8$ and simplify.

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Step 1.3.1

Divide each term in $- 8 a \geq - 4$ by $- 8$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 8 a}{- 8} \leq \frac{- 4}{- 8}$

Step 1.3.2

Simplify the left side.

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Step 1.3.2.1

Cancel the common factor of $- 8$ .

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Step 1.3.2.1.1

Cancel the common factor.

$\frac{- 8 a}{- 8} \leq \frac{- 4}{- 8}$

Step 1.3.2.1.2

Divide $a$ by $1$ .

$a \leq \frac{- 4}{- 8}$

Step 1.3.3

Simplify the right side.

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Step 1.3.3.1

Cancel the common factor of $- 4$ and $- 8$ .

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Step 1.3.3.1.1

Factor $- 4$ out of $- 4$ .

$a \leq \frac{- 4 (1)}{- 8}$

Step 1.3.3.1.2

Cancel the common factors.

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Step 1.3.3.1.2.1

Factor $- 4$ out of $- 8$ .

$a \leq \frac{- 4 \cdot 1}{- 4 \cdot 2}$

Step 1.3.3.1.2.2

Cancel the common factor.

$a \leq \frac{- 4 \cdot 1}{- 4 \cdot 2}$

Step 1.3.3.1.2.3

Rewrite the expression.

$a \leq \frac{1}{2}$

Step 2

Solve $6 a - 1 > 3 a - 10$ for $a$ .

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Step 2.1

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

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Step 2.1.1

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

$6 a - 1 - 3 a > - 10$

Step 2.1.2

Subtract $3 a$ from $6 a$ .

$3 a - 1 > - 10$

Step 2.2

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

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Step 2.2.1

Add $1$ to both sides of the inequality.

$3 a > - 10 + 1$

Step 2.2.2

Add $- 10$ and $1$ .

$3 a > - 9$

Step 2.3

Divide each term in $3 a > - 9$ by $3$ and simplify.

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Step 2.3.1

Divide each term in $3 a > - 9$ by $3$ .

$\frac{3 a}{3} > \frac{- 9}{3}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $3$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{3 a}{3} > \frac{- 9}{3}$

Step 2.3.2.1.2

Divide $a$ by $1$ .

$a > \frac{- 9}{3}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Divide $- 9$ by $3$ .

$a > - 3$

Step 3

Find the intersection of $a \leq \frac{1}{2}$ and $a > - 3$ .

$- 3 < a \leq \frac{1}{2}$

Step 4

The result can be shown in multiple forms.

Inequality Form:

$- 3 < a \leq \frac{1}{2}$

Interval Notation:

$(- 3, \frac{1}{2}]$

Step 5