Algebra Examples

$1.25 a + 3 > 0.5 a - 6$ and $2.5 a - 1 \geq 9 - 1.5 a$

Step 1

Simplify the first inequality.

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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 $0.5 a$ from both sides of the inequality.

$1.25 a + 3 - 0.5 a > - 6$ and $2.5 a - 1 \geq 9 - 1.5 a$

Step 1.1.2

Subtract $0.5 a$ from $1.25 a$ .

$0.75 a + 3 > - 6$ and $2.5 a - 1 \geq 9 - 1.5 a$

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.

$0.75 a > - 6 - 3$ and $2.5 a - 1 \geq 9 - 1.5 a$

Step 1.2.2

Subtract $3$ from $- 6$ .

$0.75 a > - 9$ and $2.5 a - 1 \geq 9 - 1.5 a$

Step 1.3

Divide each term in $0.75 a > - 9$ by $0.75$ and simplify.

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Step 1.3.1

Divide each term in $0.75 a > - 9$ by $0.75$ .

$\frac{0.75 a}{0.75} > \frac{- 9}{0.75}$ and $2.5 a - 1 \geq 9 - 1.5 a$

Step 1.3.2

Simplify the left side.

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Step 1.3.2.1

Cancel the common factor of $0.75$ .

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Step 1.3.2.1.1

Cancel the common factor.

$\frac{0.75 a}{0.75} > \frac{- 9}{0.75}$ and $2.5 a - 1 \geq 9 - 1.5 a$

Step 1.3.2.1.2

Divide $a$ by $1$ .

$a > \frac{- 9}{0.75}$ and $2.5 a - 1 \geq 9 - 1.5 a$

Step 1.3.3

Simplify the right side.

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Step 1.3.3.1

Divide $- 9$ by $0.75$ .

$a > - 12$ and $2.5 a - 1 \geq 9 - 1.5 a$

Step 2

Simplify the second inequality.

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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

Add $1.5 a$ to both sides of the inequality.

$a > - 12$ and $2.5 a - 1 + 1.5 a \geq 9$

Step 2.1.2

Add $2.5 a$ and $1.5 a$ .

$a > - 12$ and $4 a - 1 \geq 9$

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.

$a > - 12$ and $4 a \geq 9 + 1$

Step 2.2.2

Add $9$ and $1$ .

$a > - 12$ and $4 a \geq 10$

Step 2.3

Divide each term in $4 a \geq 10$ by $4$ and simplify.

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Step 2.3.1

Divide each term in $4 a \geq 10$ by $4$ .

$a > - 12$ and $\frac{4 a}{4} \geq \frac{10}{4}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $4$ .

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Step 2.3.2.1.1

Cancel the common factor.

$a > - 12$ and $\frac{4 a}{4} \geq \frac{10}{4}$

Step 2.3.2.1.2

Divide $a$ by $1$ .

$a > - 12$ and $a \geq \frac{10}{4}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Cancel the common factor of $10$ and $4$ .

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Step 2.3.3.1.1

Factor $2$ out of $10$ .

$a > - 12$ and $a \geq \frac{2 (5)}{4}$

Step 2.3.3.1.2

Cancel the common factors.

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Step 2.3.3.1.2.1

Factor $2$ out of $4$ .

$a > - 12$ and $a \geq \frac{2 \cdot 5}{2 \cdot 2}$

Step 2.3.3.1.2.2

Cancel the common factor.

$a > - 12$ and $a \geq \frac{2 \cdot 5}{2 \cdot 2}$

Step 2.3.3.1.2.3

Rewrite the expression.

$a > - 12$ and $a \geq \frac{5}{2}$

Step 3

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

$a \geq \frac{5}{2}$

Step 4

The result can be shown in multiple forms.

Inequality Form:

$a \geq \frac{5}{2}$

Interval Notation:

$[\frac{5}{2}, \infty)$

Step 5