Precalculus Examples

$4 x^{2} < 13 x - 3$

Step 1

Subtract $13 x$ from both sides of the inequality.

$4 x^{2} - 13 x < - 3$

Step 2

Convert the inequality to an equation.

$4 x^{2} - 13 x = - 3$

Step 3

Add $3$ to both sides of the equation.

$4 x^{2} - 13 x + 3 = 0$

Step 4

Factor by grouping.

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Step 4.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 4 \cdot 3 = 12$ and whose sum is $b = - 13$ .

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Step 4.1.1

Factor $- 13$ out of $- 13 x$ .

$4 x^{2} - 13 x + 3 = 0$

Step 4.1.2

Rewrite $- 13$ as $- 1$ plus $- 12$

$4 x^{2} + (- 1 - 12) x + 3 = 0$

Step 4.1.3

Apply the distributive property.

$4 x^{2} - 1 x - 12 x + 3 = 0$

Step 4.2

Factor out the greatest common factor from each group.

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Step 4.2.1

Group the first two terms and the last two terms.

$(4 x^{2} - 1 x) - 12 x + 3 = 0$

Step 4.2.2

Factor out the greatest common factor (GCF) from each group.

$x (4 x - 1) - 3 (4 x - 1) = 0$

Step 4.3

Factor the polynomial by factoring out the greatest common factor, $4 x - 1$ .

$(4 x - 1) (x - 3) = 0$

Step 5

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$4 x - 1 = 0$

$x - 3 = 0$

Step 6

Set $4 x - 1$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $4 x - 1$ equal to $0$ .

$4 x - 1 = 0$

Step 6.2

Solve $4 x - 1 = 0$ for $x$ .

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Step 6.2.1

Add $1$ to both sides of the equation.

$4 x = 1$

Step 6.2.2

Divide each term in $4 x = 1$ by $4$ and simplify.

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Step 6.2.2.1

Divide each term in $4 x = 1$ by $4$ .

$\frac{4 x}{4} = \frac{1}{4}$

Step 6.2.2.2

Simplify the left side.

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Step 6.2.2.2.1

Cancel the common factor of $4$ .

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Step 6.2.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{1}{4}$

Step 6.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{4}$

Step 7

Set $x - 3$ equal to $0$ and solve for $x$ .

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Step 7.1

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 7.2

Add $3$ to both sides of the equation.

$x = 3$

Step 8

The final solution is all the values that make $(4 x - 1) (x - 3) = 0$ true.

$x = \frac{1}{4}, 3$

Step 9

Use each root to create test intervals.

$x < \frac{1}{4}$

$\frac{1}{4} < x < 3$

$x > 3$

Step 10

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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Step 10.1

Test a value on the interval $x < \frac{1}{4}$ to see if it makes the inequality true.

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Step 10.1.1

Choose a value on the interval $x < \frac{1}{4}$ and see if this value makes the original inequality true.

$x = 0$

Step 10.1.2

Replace $x$ with $0$ in the original inequality.

$4 {(0)}^{2} < 13 (0) - 3$

Step 10.1.3

The left side $0$ is not less than the right side $- 3$ , which means that the given statement is false.

False

Step 10.2

Test a value on the interval $\frac{1}{4} < x < 3$ to see if it makes the inequality true.

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Step 10.2.1

Choose a value on the interval $\frac{1}{4} < x < 3$ and see if this value makes the original inequality true.

$x = 2$

Step 10.2.2

Replace $x$ with $2$ in the original inequality.

$4 {(2)}^{2} < 13 (2) - 3$

Step 10.2.3

The left side $16$ is less than the right side $23$ , which means that the given statement is always true.

True

Step 10.3

Test a value on the interval $x > 3$ to see if it makes the inequality true.

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Step 10.3.1

Choose a value on the interval $x > 3$ and see if this value makes the original inequality true.

$x = 6$

Step 10.3.2

Replace $x$ with $6$ in the original inequality.

$4 {(6)}^{2} < 13 (6) - 3$

Step 10.3.3

The left side $144$ is not less than the right side $75$ , which means that the given statement is false.

False

Step 10.4

Compare the intervals to determine which ones satisfy the original inequality.

$x < \frac{1}{4}$ False

$\frac{1}{4} < x < 3$ True

$x > 3$ False

$x < \frac{1}{4}$ False

$\frac{1}{4} < x < 3$ True

$x > 3$ False

Step 11

The solution consists of all of the true intervals.

$\frac{1}{4} < x < 3$

Step 12

Convert the inequality to interval notation.

$(\frac{1}{4}, 3)$

Step 13