Algebra Examples

$\sqrt{2 x + y} < \sqrt{3 x - y - 1}$

Step 1

Solve for $y$ .

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Step 1.1

To remove the radical on the left side of the inequality, square both sides of the inequality.

${\sqrt{2 x + y}}^{2} < {\sqrt{3 x - y - 1}}^{2}$

Step 1.2

Simplify each side of the inequality.

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Step 1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 x + y}$ as ${(2 x + y)}^{\frac{1}{2}}$ .

${({(2 x + y)}^{\frac{1}{2}})}^{2} < {\sqrt{3 x - y - 1}}^{2}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Simplify ${({(2 x + y)}^{\frac{1}{2}})}^{2}$ .

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Step 1.2.2.1.1

Multiply the exponents in ${({(2 x + y)}^{\frac{1}{2}})}^{2}$ .

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Step 1.2.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(2 x + y)}^{\frac{1}{2} \cdot 2} < {\sqrt{3 x - y - 1}}^{2}$

Step 1.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 1.2.2.1.1.2.1

Cancel the common factor.

${(2 x + y)}^{\frac{1}{2} \cdot 2} < {\sqrt{3 x - y - 1}}^{2}$

Step 1.2.2.1.1.2.2

Rewrite the expression.

${(2 x + y)}^{1} < {\sqrt{3 x - y - 1}}^{2}$

Step 1.2.2.1.2

Simplify.

$2 x + y < {\sqrt{3 x - y - 1}}^{2}$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Rewrite ${\sqrt{3 x - y - 1}}^{2}$ as $3 x - y - 1$ .

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Step 1.2.3.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 x - y - 1}$ as ${(3 x - y - 1)}^{\frac{1}{2}}$ .

$2 x + y < {({(3 x - y - 1)}^{\frac{1}{2}})}^{2}$

Step 1.2.3.1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$2 x + y < {(3 x - y - 1)}^{\frac{1}{2} \cdot 2}$

Step 1.2.3.1.3

Combine $\frac{1}{2}$ and $2$ .

$2 x + y < {(3 x - y - 1)}^{\frac{2}{2}}$

Step 1.2.3.1.4

Cancel the common factor of $2$ .

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Step 1.2.3.1.4.1

Cancel the common factor.

$2 x + y < {(3 x - y - 1)}^{\frac{2}{2}}$

Step 1.2.3.1.4.2

Rewrite the expression.

$2 x + y < {(3 x - y - 1)}^{1}$

Step 1.2.3.1.5

Simplify.

$2 x + y < 3 x - y - 1$

Step 1.3

Solve for $y$ .

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Step 1.3.1

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

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Step 1.3.1.1

Add $y$ to both sides of the inequality.

$2 x + y + y < 3 x - 1$

Step 1.3.1.2

Add $y$ and $y$ .

$2 x + 2 y < 3 x - 1$

Step 1.3.2

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

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Step 1.3.2.1

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

$2 y < 3 x - 1 - 2 x$

Step 1.3.2.2

Subtract $2 x$ from $3 x$ .

$2 y < x - 1$

Step 1.3.3

Divide each term in $2 y < x - 1$ by $2$ and simplify.

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Step 1.3.3.1

Divide each term in $2 y < x - 1$ by $2$ .

$\frac{2 y}{2} < \frac{x}{2} + \frac{- 1}{2}$

Step 1.3.3.2

Simplify the left side.

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Step 1.3.3.2.1

Cancel the common factor of $2$ .

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Step 1.3.3.2.1.1

Cancel the common factor.

$\frac{2 y}{2} < \frac{x}{2} + \frac{- 1}{2}$

Step 1.3.3.2.1.2

Divide $y$ by $1$ .

$y < \frac{x}{2} + \frac{- 1}{2}$

Step 1.3.3.3

Simplify the right side.

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Step 1.3.3.3.1

Move the negative in front of the fraction.

$y < \frac{x}{2} - \frac{1}{2}$

Step 2

Find the slope and the y-intercept for the boundary line.

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Step 2.1

Rewrite in slope-intercept form.

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Step 2.1.1

The slope-intercept form is $y = m x + b$ , where $m$ is the slope and $b$ is the y-intercept.

$y = m x + b$

Step 2.1.2

Reorder terms.

$y < \frac{1}{2} x - \frac{1}{2}$

Step 2.2

Use the slope-intercept form to find the slope and y-intercept.

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Step 2.2.1

Find the values of $m$ and $b$ using the form $y = m x + b$ .

$m = \frac{1}{2}$

$b = - \frac{1}{2}$

Step 2.2.2

The slope of the line is the value of $m$ , and the y-intercept is the value of $b$ .

Slope: $\frac{1}{2}$

y-intercept: $(0, - \frac{1}{2})$

Slope: $\frac{1}{2}$

y-intercept: $(0, - \frac{1}{2})$

Slope: $\frac{1}{2}$

y-intercept: $(0, - \frac{1}{2})$

Step 3

Graph a dashed line, then shade the area below the boundary line since $y$ is less than $\frac{x}{2} - \frac{1}{2}$ .

$y < \frac{x}{2} - \frac{1}{2}$

Step 4