Chapter 1

The imaginary number \(i\) has the property that \(i^{2}=\) _________

The equation \(|X|=3\) has the two solutions____and____

Fill in the blank with an appropriate inequality sign. (a) If \(x<5,\) then \(x-3 ________\) (b) If \(x \leq 5,\) then 3\(x ________ 15\) (c) If \(x \geq 2,\) then \(-3x _______ -6\) (d) If \(x<-2,\) then \(-x _________ 2\)

(a) The solutions of the equation \(x^{2}(x-4)=0\) are _____ (b) To solve the equation \(x^{3}-4 x^{2}=0,\) we _____ the left-hand side.

The Quadratic Formula gives us the solutions of the equation \(a x^{2}+b x+c=0\) (a) State the Quadratic Formula: \(x=\) _____. (b) In the equation \(\frac{1}{2} x^{2}-x-4=0, a=\) _____, \(b=\) _____, and \(c=\) _____. So the solution of the equation is \(x=\) _____.

Explain in your own words what it means for an equation to model a real-world situation, and give an example.

Which of the following equations are linear? (a) \(\frac{X}{2}+2 x=10\) (b) \(\frac{2}{x}-2 x=1\) (c) \(x+7=5-3 x\)

For the complex number \(3+4 i\) the real part is _________ the imaginary part is ____________

The solution of the inequality \(|x| \leq 3\) is the interval________

Solve the equation \(\sqrt{2 x}+x=0\) by doing the following steps. (a) Isolate the radical: _____ (b) Square both sides: _____ (c) The solutions of the resulting quadratic equation are _____ (d) The solution(s) that satisfy the original equation are _____

Explain how you would use each method to solve the equation \(X^{2}-4 x-5=0\) (a) By factoring: _____. (b) By completing the square: _____. (c) By using the Quadratic Formula: _____.

Explain why each of the following equations is not linear. (a) \(x(x+1)=6\) (b) \(\quad \sqrt{x+2}=X\) (c) \(3 x^{2}-2 x-1=0\)

(a) The complex conjugate of \(3+4 i\) is \(\overline{3+4 i}=\) ________ (b) \((3+4 i)(\overline{3+4 i})=\) ________

The solution of the inequality \(|x| \geq 3\) is a union of two intervals________\(\cup\)_______

Let \(S=\left\\{-2,-1,0, \frac{1}{2}, 1, \sqrt{2}, 2,4\right\\} .\) Determine which elements of \(S\) satisfy the inequality. $$ x-3>0 $$

Give a formula for the area of the geometric figure. (a) A square of side \(x . \quad A=\)_______ (b) A rectangle of length \(l\) and width \(w : \quad A=\)_________ (c) A circle of radius \(r . \quad A=\)_________

True or false? (a) Adding the same number to each side of an equation always gives an equivalent equation. (b) Multiplying each side of an equation by the same number always gives an equivalent equation. (c) Squaring each side of an equation always gives an equivalent equation.

If \(3+4 i\) is a solution of a quadratic equation with real coefficients, then ________ is also a solution of the equation.

(a) The set of all points on the real line whose distance from zero is less than 3 can be described by the absolute value inequality \(|x|\)_______ (b) The set of all points on the real line whose distance from zero is greater than 3 can be described by the absolute value inequality \(|x|\)_________

Let \(S=\left\\{-2,-1,0, \frac{1}{2}, 1, \sqrt{2}, 2,4\right\\} .\) Determine which elements of \(S\) satisfy the inequality. $$ x+1<2 $$

Make up quadratic equations that have the following number of solutions: Two solutions: _____. One solution: _____. No solution: _____.

Balsamic vinegar contains 5\(\%\) acetic acid, so a \(32-\) oz bottle of balsamic vinegar contains ________ounces of acetic acid.

To solve the equation \(x^{3}=125,\) we take the ________ root of each side. So the solution is \(x=\) _________.

Find the real and imaginary parts of the complex number. $$ 5-7 i $$

\(5-22=\) Solve the equation. $$ |4 x|=24 $$

Let \(S=\left\\{-2,-1,0, \frac{1}{2}, 1, \sqrt{2}, 2,4\right\\} .\) Determine which elements of \(S\) satisfy the inequality. $$ 3-2 x \leq \frac{1}{2} $$

Solve the equation by factoring. $$ x^{2}+x=12 $$

\(5-60\) Find all real solutions of the equation. $$ x^{3}=16 x $$

A painter paints a wall in \(x\) hours, so the fraction of the wall that she paints in 1 hour is_____

Determine whether the given value is a solution of the equation. \(4 x+7=9 x-3\) (a) \(x=-2 \quad\) (b) \(x=2\)

Find the real and imaginary parts of the complex number. $$ -6+4 i $$

\(5-22=\) Solve the equation. $$ |6 x|=15 $$

Let \(S=\left\\{-2,-1,0, \frac{1}{2}, 1, \sqrt{2}, 2,4\right\\} .\) Determine which elements of \(S\) satisfy the inequality. $$ 2 x-1 \geq x $$

Solve the equation by factoring. $$ x^{2}+3 x=4 $$

\(5-60\) Find all real solutions of the equation. $$ x^{5}=27 x^{2} $$

The formula \(d=r t\) models the distance \(d\) traveled by an object moving at the constant rate \(r\) in time \(t\) . Find formulas for the following quantities.

Determine whether the given value is a solution of the equation. \(2-5 x=8+x\) (a) \(x=-1 \quad\) (b) \(x=1\)

Find the real and imaginary parts of the complex number. $$ \frac{-2-5 i}{3} $$

\(5-22=\) Solve the equation. $$ 5|x|+3=28 $$

Let \(S=\left\\{-2,-1,0, \frac{1}{2}, 1, \sqrt{2}, 2,4\right\\} .\) Determine which elements of \(S\) satisfy the inequality. $$ 1<2 x-4 \leq 7 $$

Solve the equation by factoring. $$ x^{2}-7 x+12=0 $$

\(5-60\) Find all real solutions of the equation. $$ x^{6}-81 x^{2}=0 $$

\(7-18 \cdot\) Express the given quantity in terms of the indicated variable. The sum of three consecutive integers; \(\quad n=\) first integer of the three

Determine whether the given value is a solution of the equation. \(1-[2-(3-x)]=4 x-(6+x)\) (a) \(x=2 \quad\) (b) \(x=4\)

Find the real and imaginary parts of the complex number. $$ \frac{4+7 i}{2} $$

\(5-22=\) Solve the equation. $$ \frac{1}{2}|x|-7=2 $$

Let \(S=\left\\{-2,-1,0, \frac{1}{2}, 1, \sqrt{2}, 2,4\right\\} .\) Determine which elements of \(S\) satisfy the inequality. $$ -2 \leq 3-x<2 $$

Solve the equation by factoring. $$ x^{2}+8 x+12=0 $$

\(5-60\) Find all real solutions of the equation. $$ x^{5}-16 x=0 $$

\(7-18 \cdot\) Express the given quantity in terms of the indicated variable. The sum of three consecutive integers; \(\quad n=\) middle integer of the three