Solve 9z^2≤10z-1

Question

Solve the inequality

Solve the inequality by testing the values in the interval

Solve the inequality by separating into cases

Solve for z

\frac{1}{9} \leq z \leq 1

Alternative Form

z \in [\frac{1}{9}, 1]

Evaluate

9 z^{2} \leq 10 z - 1

Move the expression to the left side

9 z^{2} - (10 z - 1) \leq 0

If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses

9 z^{2} - 10 z + 1 \leq 0

Rewrite the expression

9 z^{2} - 10 z + 1 = 0

Factor the expression

More Steps

(z - 1) (9 z - 1) = 0

When the product of factors equals 0,at least one factor is 0

z - 1 = 0 9 z - 1 = 0

Solve the equation for z

More Steps

z = 1 9 z - 1 = 0

Solve the equation for z

More Steps

z = 1 z = \frac{1}{9}

Determine the test intervals using the critical values

z < \frac{1}{9} \frac{1}{9} < z < 1 z > 1

Choose a value form each interval

z_{1} = - 1 z_{2} = \frac{5}{9} z_{3} = 2

To determine if z < \frac{1}{9} is the solution to the inequality,test if the chosen value z = - 1 satisfies the initial inequality

More Steps

z < \frac{1}{9} is not a solution z_{2} = \frac{5}{9} z_{3} = 2

To determine if \frac{1}{9} < z < 1 is the solution to the inequality,test if the chosen value z = \frac{5}{9} satisfies the initial inequality

More Steps

z < \frac{1}{9} is not a solution \frac{1}{9} < z < 1 is the solution z_{3} = 2

To determine if z > 1 is the solution to the inequality,test if the chosen value z = 2 satisfies the initial inequality

More Steps

z < \frac{1}{9} is not a solution \frac{1}{9} < z < 1 is the solution z > 1 is not a solution

The original inequality is a nonstrict inequality,so include the critical value in the solution

\frac{1}{9} \leq z \leq 1 is the solution

Solution

\frac{1}{9} \leq z \leq 1

Alternative Form

z \in [\frac{1}{9}, 1]

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