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Precalculus
Algebra Review

1
The Real Number System

Problem  1
practice icon

What property does each of the following equations illustrate?

  1. \(4+6=6+4\)

  2. \(3(5-8)=3(5)-3(8)\)

  3. \(2\cdot (3\cdot 5)=(2\cdot 3)\cdot 5\)

  4. \(4\cdot 1=4\)

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Mr. Hampton cc
Mr. Hampton
Molly S. cc
Molly S.
Gordon cc
Gordon
Saba cc
Saba
Problem  2

Graph each of the following intervals on the number line and give a verbal description.

  1. \((-2, 4]\)

  2. \((-\infty, 4]\)

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Mr. Hampton cc
Mr. Hampton
Molly S. cc
Molly S.
Gordon cc
Gordon
Saba cc
Saba
Problem  3

Give the corresponding interval notation for the graph of an inequality.

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Mr. Hampton cc
Mr. Hampton
Molly S. cc
Molly S.
Gordon cc
Gordon
Saba cc
Saba
Problem  4
practice icon

Evaluate the following.

  1. \(\lvert -3\rvert\)

  2. \(-4-\lvert 4.5\rvert\)

  3. \(\lvert -4+9\rvert +3\)

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Mr. Hampton cc
Mr. Hampton
Molly S. cc
Molly S.
Gordon cc
Gordon
Saba cc
Saba
Problem  5

Find the distance between \(-6\) and \(4\) on the real number line.

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Mr. Hampton cc
Mr. Hampton
Molly S. cc
Molly S.
Gordon cc
Gordon
Saba cc
Saba
Problem  6
practice icon

Simplify each expression.

  1. \(3^2+2(-3+7)\)

  2. \(\dfrac{3^2+1}{4-7}\)

  3. \(6+3(5-(-3^2+2))\)

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Mr. Hampton cc
Mr. Hampton
Gordon cc
Gordon
Saba cc
Saba
Problem  7
practice icon

Evaluate the following expression on your calculator. Check the calculator output with a hand calculation.

  1. \(4+10\div 5+2\)

  2. \(\dfrac{4+10}{5+2}\)

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Mr. Hampton cc
Mr. Hampton
Molly S. cc
Molly S.
Gordon cc
Gordon
Saba cc
Saba

2
Exponents, Roots, and Radicals

Problem  1
practice icon

Write each expression using positive exponents.

  1. \(4^{-2}\)

  2. \(1.45^{-3}\)

  3. \(\dfrac{x^{-4}}{y^5}\), \(y\neq 0\)

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Mr. Hampton cc
Mr. Hampton
Molly S. cc
Molly S.
Gordon cc
Gordon
Saba cc
Saba
Problem  2
practice icon

Simplify each expression and write it using positive exponents. Assume that variables represent nonzero real numbers.

  1. \(\dfrac{16x^{10}y^4}{4x^{10}y^8}\)

  2. \(\dfrac{\left(3x^{-2}\right)^3}{x^{-5}}\)

  3. \(\left(\dfrac{54s^2t^{-3}}{6zt}\right)^{-2}\)

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Mr. Hampton cc
Mr. Hampton
Molly S. cc
Molly S.
Gordon cc
Gordon
Saba cc
Saba
Problem  3
practice icon

Express each of the following numbers in scientific notation.

  1. \(328.5\)

  2. \(4.69\)

  3. \(0.00712\)

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Mr. Hampton cc
Mr. Hampton
Molly S. cc
Molly S.
Gordon cc
Gordon
Saba cc
Saba
Problem  4
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Write the following numbers in decimal form.

  1. \(2.1\times 10^5\)

  2. \(3.47\times 10^{-3}\)

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Mr. Hampton cc
Mr. Hampton
Molly S. cc
Molly S.
Gordon cc
Gordon
Saba cc
Saba
Problem  5
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In 2012, Japan had a population os \(1.27\times 10^8\) and a land area of \(3.64\times 10^5\) square kilometers. What is the population density of Japan, that is, the number of people per square kilometer of land? Express your answer in scientific notation.

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Mr. Hampton cc
Mr. Hampton
Molly S. cc
Molly S.
Gordon cc
Gordon
Octabio cc espanol spanish
Octabio
Problem  6
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Determine \(\sqrt[3]{64}\), \(\sqrt[4]{64}\), and \(\sqrt[5]{-32}\).

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Mr. Hampton cc
Mr. Hampton
Molly S. cc
Molly S.
Gordon cc
Gordon
Saba cc
Saba
Problem  7
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Rationalize the denominator of each expression.

  1. \(\sqrt{\dfrac{18}{5}}\)

  2. \(\dfrac{\sqrt[3]{10}}{\sqrt[3]{3}}\)

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Mr. Hampton cc
Mr. Hampton
Molly S. cc
Molly S.
Gordon cc
Gordon
Saba cc
Saba
Problem  8
practice icon

Simplify the following.

  1. \(\sqrt{75}\)

  2. \(\sqrt[3]{\dfrac{125}{108}}\)

  3. \(\sqrt{x^5y^7}\), \(x,y>0\)

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Mr. Hampton cc
Mr. Hampton
Molly S. cc
Molly S.
Gordon cc
Gordon
Octabio cc espanol spanish
Octabio
Problem  9
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Rationalize the denominator.
\(\dfrac{5}{4-\sqrt{2}}\)

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Mr. Hampton cc
Mr. Hampton
Molly S. cc
Molly S.
Gordon cc
Gordon
Saba cc
Saba
Problem  10
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Simplify the following radical expressions. Assume \(x\geq0\).

  1. \(\sqrt{48}+\sqrt{27}-\sqrt{12}\)

  2. \(\left(3+\sqrt{x}\right)\left(4-2\sqrt{x}\right)\)

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Mr. Hampton cc
Mr. Hampton
Molly S. cc
Molly S.
Gordon cc
Gordon
Saba cc
Saba
Problem  11
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Evaluate the following.

  1. \(27^\frac{1}{3}\)

  2. \(-36^\frac{1}{2}\)

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Mr. Hampton cc
Mr. Hampton
Molly S. cc
Molly S.
Gordon cc
Gordon
Saba cc
Saba
Problem  12
practice icon

Evaluate the following expressions without using a calculator.

  1. \((4)^\frac{1}{2}\)

  2. \(\left(\dfrac{1}{3}\right)^{-2}\)

  3. \((23.1)^0\)

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Mr. Hampton cc
Mr. Hampton
Molly S. cc
Molly S.
Gordon cc
Gordon
Octabio cc espanol spanish
Octabio
Problem  13
practice icon

Simplify and write positive exponents.

  1. \((25)^\frac{1}{2}\)

  2. \(\left(16s^\frac{4}{3}t^{-3}\right)^\frac{3}{2}\), \(s,t>0\)

  3. \(\dfrac{\left(x^{-\frac{2}{3}}y^{-3}\right)^3}{y^{-\frac{1}{2}}x^2}\), \(x,y>0\)

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Mr. Hampton cc
Mr. Hampton
Lauren cc
Lauren
Gordon cc
Gordon
Saba cc
Saba

3
Polynomials and Factoring

Problem  1

Write the following polynomial in descending order and find its degree, terms, coefficients, and constant term. \[-4x^2+3x^5-2x-7\]

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Mr. Hampton cc
Mr. Hampton
Molly S. cc
Molly S.
Gordon cc
Gordon
Saba cc
Saba
Problem  2
practice icon

Add or subtract each of the following.

  1. \(\left(3x^3+2x^2-5x+7\right)+\left(x^3-x^2+5x-2\right)\)

  2. \(\left(s^4+\dfrac{3}{4}s^2\right)-\left(s^4-s^2\right)\)

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Mr. Hampton cc
Mr. Hampton
Molly S. cc
Molly S.
Gordon cc
Gordon
Octabio cc espanol spanish
Octabio
Problem  3
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Multiply \(\left(-2x^2\right)\left(4x^7\right)\)

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Mr. Hampton cc
Mr. Hampton
Molly S. cc
Molly S.
Gordon cc
Gordon
Saba cc
Saba
Problem  4
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Multiply \((3x+2)(-2x-3)\)

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Mr. Hampton cc
Mr. Hampton
Molly S. cc
Molly S.
Gordon cc
Gordon
Saba cc
Saba
Problem  5
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Multiply \((-7x+4)(5x-1)\)

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Mr. Hampton cc
Mr. Hampton
Molly S. cc
Molly S.
Gordon cc
Gordon
Octabio cc espanol spanish
Octabio
Problem  6
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Multiply \((4y^3-3y+1)(y-2)\)

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Mr. Hampton cc
Mr. Hampton
Molly S. cc
Molly S.
Gordon cc
Gordon
Saba cc
Saba
Problem  7

Factor the greatest common factor from each of the following.

  1. \(3x^4+9x^3+18x^2\)

  2. \(-8y^2-6y+4\)

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Mr. Hampton cc
Mr. Hampton
Lauren cc
Lauren
Gordon cc
Gordon
Saba cc
Saba
Problem  8
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Factor \(x^3-x^2+2x-2\) by grouping.

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Mr. Hampton cc
Mr. Hampton
Shelby cc
Shelby
Gordon cc
Gordon
Saba cc
Saba
Problem  9
practice icon

Factor each of the following

  1. \(x^2-2x-8\)

  2. \(8x^3-10x^2-12x\)

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Mr. Hampton cc
Mr. Hampton
Shelby cc
Shelby
Gordon cc
Gordon
Saba cc
Saba
Problem  10

Factor using one of the special factorization patterns.

  1. \(27x^3-64\)

  2. \(8x^2+32x+32\)

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Mr. Hampton
Mr. Hampton
Shelby cc
Shelby
Gordon cc
Gordon
Saba cc
Saba

4
Rational Expressions

Problem  1
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For what value of \(x\) is the following rational expression defined? \[\frac{x+1}{(x-3)(x-5)}\]

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Mr. Hampton cc
Mr. Hampton
Lauren cc
Lauren
Gordon cc
Gordon
Nathan cc
Nathan
Problem  2
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Simplify: \(\dfrac{x^2-2x+1}{3x^2-4x+1}\)

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Mr. Hampton cc
Mr. Hampton
Lauren cc
Lauren
Gordon cc
Gordon
Octabio cc espanol spanish
Octabio
Problem  3

Multiply the following and express your answer in lowest terms. For what values of the variable is the expression meaningful?

  1. \(\dfrac{3a}{8}\cdot\dfrac{24}{6a^3}\)

  2. \(\dfrac{x^2+x-6}{x^2-4}\cdot\dfrac{(x+2)^2}{\left(x^2+9\right)}\)

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Saba cc
Saba
Lauren cc
Lauren
Breylor cc
Breylor
Julieta cc espanol spanish
Julieta
Problem  4
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Divide and simplify:
\(\dfrac{3x^2-5x-2}{x^2-4x+4}\div\dfrac{9x^2-1}{x+5}\)

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Mr. Hampton cc
Mr. Hampton
Lauren cc
Lauren
Breylor cc
Breylor
Julieta cc espanol spanish
Julieta
Problem  5

Add or subtract the following and express your answer in lowest terms.

  1. \(\dfrac{x+4}{3x+6}+\dfrac{2x+1}{x^2+7x+10}\)

  2. \(\dfrac{3x}{4-2x}-\dfrac{x+5}{x^2-4}\)

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Mr. Hampton cc
Mr. Hampton
Lauren cc
Lauren
Gordon cc
Gordon
Octabio cc espanol spanish
Octabio
Problem  6
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Simplify \(\dfrac{\dfrac{x^2-4}{2x+1}}{\dfrac{x^2+x-6}{x-1}}\)

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Mr. Hampton cc
Mr. Hampton
Lauren cc
Lauren
Gordon cc
Gordon
Julieta cc espanol spanish
Julieta
Problem  7
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Simplify \(\dfrac{\dfrac{1}{x}+\dfrac{1}{xy}}{\dfrac{3}{y^2}+\dfrac{1}{y}}\)

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Mr. Hampton cc
Mr. Hampton
Lauren cc
Lauren
Gordon cc
Gordon
Julieta
Julieta

5
Linear and Quadratic Equations

Problem  1
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Solve the following equation for \(x\).
\(3(x+2)-2=4x\)

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Saba cc
Saba
Lauren cc
Lauren
Gordon cc
Gordon
Julieta cc espanol spanish
Julieta
Problem  2
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Solve the equation:

\(\dfrac{x+5}{2}+\dfrac{2x-1}{5}=5\)

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Mr. Hampton cc
Mr. Hampton
Lauren cc
Lauren
Breylor cc
Breylor
Octabio cc espanol spanish
Octabio
Problem  3
practice icon

Solve the equation:

\(0.3(x+2)-0.02x=0.5\)

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Mr. Hampton cc
Mr. Hampton
Lauren cc
Lauren
Breylor cc
Breylor
Julieta cc espanol spanish
Julieta
Problem  4
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The perimeter of a rectangular fence is \(15\) feet. Write the width of the fence in terms of the length.

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Mr. Hampton cc
Mr. Hampton
Lauren cc
Lauren
Breylor cc
Breylor
Julieta cc espanol spanish
Julieta
Problem  5
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Solve \(2x^2-7x+3=0\) by factoring.

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Mr. Hampton cc
Mr. Hampton
Lauren cc
Lauren
Gordon cc
Gordon
Octabio cc espanol spanish
Octabio
Problem  6
practice icon

Solve \(-3x^2+9=0\)

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Mr. Hampton cc
Mr. Hampton
Lauren cc
Lauren
Breylor cc
Breylor
Julieta cc espanol spanish
Julieta
Problem  7
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Solve the equation \(3x^2-6x-1=0\) by using the technique of completing the square.

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Mr. Hampton cc
Mr. Hampton
Lauren cc
Lauren
Gordon cc
Gordon
Julieta cc espanol spanish
Julieta
Problem  8
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Solve the equation \(-4x^2+3x+\dfrac{1}{2}=0\) by using the quadratic formula.

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Mr. Hampton cc
Mr. Hampton
Lauren cc
Lauren
Gordon cc
Gordon
Julieta cc espanol spanish
Julieta
Problem  9
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Use the quadratic formula to find the real solutions of the equation \(-2t^2+3t=5\). Find the value of the discriminant.

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Mr. Hampton cc
Mr. Hampton
Lauren cc
Lauren
Gordon cc
Gordon
Octabio cc espanol spanish
Octabio
Problem  10
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The height of a ball after being thrown vertically upward from a point \(80\) feet above the ground with a velocity of \(40\) feet per second is given by \(h=-16t^2+40t+80\), where \(t\) is the time in seconds since the ball was thrown and \(h\) is in feet.

  1. When will the ball be \(50\) feet above the ground?

  2. When will the ball reach the ground?

  3. For what values of \(t\) does this problem make sense (from a physical sense)?

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Mr. Hampton cc
Mr. Hampton
Lauren cc
Lauren
Gordon cc
Gordon
Julieta cc espanol spanish
Julieta

6
Linear Inequalities

Problem  1
practice icon

Solve \(x-4>-2x+2\)

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Mr. Hampton cc
Mr. Hampton
Lauren cc
Lauren
Gordon cc
Gordon
Breylor cc
Breylor
Problem  2

Solve the following inequalities:

  1. \(2x+\dfrac{5}{2}>3x-6\)

  2. \(-4\leq 3x-2<7\)

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Mr. Hampton cc
Mr. Hampton
Lauren cc
Lauren
Gordon cc
Gordon
Julieta cc espanol spanish
Julieta
Problem  3
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Alicia has a total of \(\$1\text{,}000\) to spend on a new computer system. If the sales tax is \(8\%\), what is the retail price range of computers that she should consider?

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Mr. Hampton cc
Mr. Hampton
Lauren cc
Lauren
Gordon cc
Gordon
Octabio cc espanol spanish
Octabio
Problem  4

The Verizon phone company in New Jersey has the following two plans for local toll calls:

Plan A charges \(\$4.00\) per month plus 8 cents per minute for every local toll call.

Plan B charges a flat rate of \(\$20\) per month for local toll calls regardless of the number of minutes of use.

a. Express the monthly cost for Plan A in terms of the number of minutes used.

b. Express the monthly cost for Plan B in terms of the number of minutes used.

c. How many minutes would you have to use per month for Plan B to be cheaper than Plan A?

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Mr. Hampton
Mr. Hampton
Lauren
Lauren
Gordon cc
Gordon
Problem  5
practice icon

To operate a gourmet coffee booth in a shopping mall, it costs \(\$500\) (the fixed cost) plus \(\$6\) for each pound of coffee bought at wholesale price. The coffee is then sold to customers for \(\$10\) per pound.

  1. Find an expression for the operating cost of selling \(q\) pounds of coffee.

  2. Find an expression for the revenue earned by selling \(q\) pounds of coffee.

  3. Find the break-even point.

  4. How many pounds of coffee must be sold for the revenue to be greater than the total cost?

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Mr. Hampton cc
Mr. Hampton
Lauren cc
Lauren
Gordon cc
Gordon
Julieta cc espanol spanish
Julieta

7
Equations and Inequalities Involving Absolute Value

Problem  1

Solve the following equations:

  1. \(\lvert 2x-3\rvert =7\)

  2. \(\lvert x \rvert = -3\)

  3. \(-\lvert 3x+1\rvert -3=-8\)

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Mr. Hampton cc
Mr. Hampton
Lauren cc
Lauren
Gordon cc
Gordon
Julieta cc espanol spanish
Julieta
Problem  2

Solve the following inequalities and indicate the solution set on a number line.

  1. \(\lvert 2x-3\rvert >7\)

  2. \(\left\lvert -\dfrac{2}{3}x+4\right\rvert \leq 57\)

  3. \(-4+\lvert 3-x\rvert >5\)

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Mr. Hampton cc
Mr. Hampton
Lauren cc
Lauren
Gordon cc
Gordon
Julieta cc espanol spanish
Julieta
Problem  3

Graph the following on a number line and write each set using an absolute value inequality.

  1. The set of all \(x\) whose distance from \(4\) is less than \(5\)

  2. The set of all \(x\) whose distance from \(4\) is greater than \(5\)

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Mr. Hampton cc
Mr. Hampton
Lauren cc
Lauren
Breylor cc
Breylor
Octabio cc espanol spanish
Octabio
Problem  4

A thermometer measures temperature with an uncertainty of \(0.25^\circ\text{F}\). If a person’s body temperature is measured at \(98.3^\circ\text{F}\), use absolute value notation to write an inequality for the range of possible body temperatures.

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Mr. Hampton cc
Mr. Hampton
Lauren cc
Lauren
Gordon cc
Gordon
Julieta cc espanol spanish
Julieta

8
Other Types of Equations

Problem  1
practice icon

Solve the equation \(3x^4+5x^2-2=0\) for real values of \(x\).

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Mr. Hampton cc
Mr. Hampton
Lauren cc
Lauren
Gordon cc
Gordon
Julieta cc espanol spanish
Julieta
Problem  2

Solve the equation \(t^6-t^3=2\) for real values of \(t\).

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Mr. Hampton cc
Mr. Hampton
Lauren cc
Lauren
Gordon cc
Gordon
Breylor cc
Breylor
Problem  3

Solve \(\dfrac{1}{x}=\dfrac{2}{x-2}+3\)

Choose instructor to watch:
Mr. Hampton cc
Mr. Hampton
Lauren cc
Lauren
Gordon cc
Gordon
Julieta cc espanol spanish
Julieta
Problem  4
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Solve \(\sqrt{3x+1}+2=x-1\)

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Mr. Hampton cc
Mr. Hampton
Lauren cc
Lauren
Gordon cc
Gordon
Julieta cc espanol spanish
Julieta
Problem  5
practice icon

Solve \(\sqrt{3x+1}-\sqrt{x+4}=1\)

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Mr. Hampton cc
Mr. Hampton
Gordon cc
Gordon
Octabio cc espanol spanish
Octabio
Problem  6

A theatre club arranged for a chartered bus trip to a play at a cost of \(\$350\). To lower costs, \(10\) nonmembers were invited to join the trip. The bus fare per person then decreased by \(\$4\). How many theatre club members were going on the trip?

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Mr. Hampton cc
Mr. Hampton
Lauren cc
Lauren
Gordon cc
Gordon
Julieta cc espanol spanish
Julieta
Problem  7

Jennifer is standing on one side of a river that is \(3\) kilometers wide. Her bus is located on the opposite side of the river. Jennifer plans to cross the river by rowboat and then jog the rest of the way to reach the bus, which is \(10\) kilometers along the river from point \(B\) directly across the river from her current location (point \(A\)). If she can row \(5\) kilometers per hour and jog \(7\) kilometers per hour, at which point on the other side of the river should she dock her boat so that it will take her a total of exactly two hours to reach the bus? Assume that Jennifer’s path on each leg of the trip is a straight line, and that there is no river current or wind speed.

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Mr. Hampton cc
Mr. Hampton
Lauren cc
Lauren
Gordon cc
Gordon
Julieta cc espanol spanish
Julieta