Section 1
Linear Equations and Rational Equations
Fill in the blanks.If a number satisfies an equation, it is called a ________or a _______________of the equation.
If an equation is true for all values of its variable, it is called an _______________.
A contradiction is an equation that is true for ___________ values of its variable.
A ______________________ equation is true for some values of its variable and is not true for others.
An equation of the form $a x+b=0$ is called a __________________ equation.
If an equation contains rational expressions, it is called a __________________equation.
A conditional linear equation has ________________ root.
The ___________________ of a fraction can never be 0
Each quantity represents a real mumber. Find any restrictions on $x .$.$$x+3=1$$
Each quantity represents a real mumber. Find any restrictions on $x .$.$$\frac{1}{2} x-7=14$$
Each quantity represents a real mumber. Find any restrictions on $x .$.$$\frac{1}{x}=12$$
Each quantity represents a real mumber. Find any restrictions on $x .$.$$\frac{3}{x-2}=9 x$$
Each quantity represents a real mumber. Find any restrictions on $x .$.$$\frac{8}{x-3}=\frac{5}{x+2}$$
Each quantity represents a real mumber. Find any restrictions on $x .$.$$\frac{x}{x-3}=-\frac{4}{x+4}$$
Each quantity represents a real mumber. Find any restrictions on $x .$.$$\frac{1}{x-3}=\frac{5 x}{x^{2}-16}$$
Each quantity represents a real mumber. Find any restrictions on $x .$.$$\frac{1}{x^{2}-3 x-4}=\frac{5}{x}+2$$
Solve each equation, if possible. Classify each one as an identity, a conditional equation, or a contradiction.$$2 x+5=15$$
Solve each equation, if possible. Classify each one as an identity, a conditional equation, or a contradiction.$$3 x+2=x+8$$
Solve each equation, if possible. Classify each one as an identity, a conditional equation, or a contradiction.$$2(n+2)-5=2 n$$
Solve each equation, if possible. Classify each one as an identity, a conditional equation, or a contradiction.$$3(m+2)=2(m+3)+m$$
Solve each equation, if possible. Classify each one as an identity, a conditional equation, or a contradiction.$$\frac{x+7}{2}=7$$
Solve each equation, if possible. Classify each one as an identity, a conditional equation, or a contradiction.$$\frac{x}{2}-7=14$$
Solve each equation, if possible. Classify each one as an identity, a conditional equation, or a contradiction.$$2(a+1)=3(a-2)-a$$
Solve each equation, if possible. Classify each one as an identity, a conditional equation, or a contradiction.$$x^{2}=(x+4)(x-4)+16$$
Solve each equation, if possible. Classify each one as an identity, a conditional equation, or a contradiction.$$3(x-3)=\frac{6 x-18}{2}$$
Solve each equation, if possible. Classify each one as an identity, a conditional equation, or a contradiction.$$x(x+2)=(x+1)^{2}$$
Solve each equation, if possible. Classify each one as an identity, a conditional equation, or a contradiction.$$\frac{3}{b-3}=1$$
Solve each equation, if possible. Classify each one as an identity, a conditional equation, or a contradiction.$$x^{2}-8 x+15=(x-3)(x+5)$$
Solve each equation, if possible. Classify each one as an identity, a conditional equation, or a contradiction.$$2 x^{2}+5 x-3=(2 x-1)(x+3)$$
Solve each equation, if possible. Classify each one as an identity, a conditional equation, or a contradiction.$$2 x^{2}+5 x-3=2 x\left(x+\frac{19}{2}\right)$$
Solve each equation. If an equation has no solution, so indicate.$$2 x+7=10-x$$
Solve each equation. If an equation has no solution, so indicate.$$9 a-3=15+3 a$$
Solve each equation. If an equation has no solution, so indicate.$$5(x-2)=2(x+4)$$
Solve each equation. If an equation has no solution, so indicate.$$5(r-4)=-5(r-4)$$
Solve each equation. If an equation has no solution, so indicate.$$7(2 x+5)-6(x+8)=7$$
Solve each equation. If an equation has no solution, so indicate.$$6(x-5)-4(x+2)=-1$$
Solve each equation. If an equation has no solution, so indicate.$$\frac{5}{3} z-8=7$$
Solve each equation. If an equation has no solution, so indicate.$$\frac{4}{3} y+12=-4$$
Solve each equation. If an equation has no solution, so indicate.$$\frac{z}{5}+2=4$$
Solve each equation. If an equation has no solution, so indicate.$$\frac{3 p}{7}-p=-4$$
Solve each equation. If an equation has no solution, so indicate.$$\frac{3 x-2}{3}=2 x+\frac{7}{3}$$
Solve each equation. If an equation has no solution, so indicate.$$\frac{7}{2} x+5=x+\frac{15}{2}$$
Solve each equation. If an equation has no solution, so indicate.$$\frac{3 x+1}{20}=\frac{1}{2}$$
Solve each equation. If an equation has no solution, so indicate.$$2 x-\frac{7}{6}+\frac{x}{6}=\frac{4 x+3}{6}$$
Solve each equation. If an equation has no solution, so indicate.$$\frac{3+x}{3}+\frac{x+7}{2}=4 x+1$$
Solve each equation. If an equation has no solution, so indicate.$$2(2 x+1)-\frac{3 x}{2}=\frac{-3(4+x)}{2}$$
Solve each equation. If an equation has no solution, so indicate.$$\frac{3}{2}(3 x-2)-10 x-4=0$$
Solve each equation. If an equation has no solution, so indicate.$$\frac{a(a-3)+5}{7}=\frac{(a-1)^{2}}{7}$$
Solve each equation. If an equation has no solution, so indicate.$$\frac{(y+2)^{2}}{3}=y+2+\frac{y^{2}}{3}$$
Solve each equation. If an equation has no solution, so indicate.$$(t+1)(t-1)=(t+2)(t-3)+4$$
Solve each equation. If an equation has no solution, so indicate.$$x(x+2)=(x+1)^{2}-1$$
Solve each equation. If an equation has no solution, so indicate.$$(x-2)(x-3)=(x+3)(x+4)$$
Solve each equation. If an equation has no solution, so indicate.$$2(s+2)+(s+3)^{2}=s(s+5)+2\left(\frac{17}{2}+s\right)$$
Solve each equation. If an equation has no solution, so indicate.$$\frac{3}{x}+\frac{1}{2}=\frac{4}{x}$$
Solve each equation. If an equation has no solution, so indicate.$$\frac{2}{x+1}+\frac{1}{3}=\frac{1}{x+1}$$
Solve each equation. If an equation has no solution, so indicate.$$\frac{3}{x-2}+\frac{1}{x}=\frac{3}{x-2}$$
Solve each equation. If an equation has no solution, so indicate.$$\frac{9 t+6}{t(t+3)}=\frac{7}{t+3}$$
Solve each equation. If an equation has no solution, so indicate.$$x+\frac{2(-2 x+1)}{3 x+5}=\frac{3 x^{2}}{3 x+5}$$
Solve each equation. If an equation has no solution, so indicate.$$\frac{2}{(a-7)(a+2)}=\frac{4}{(a+3)(a+2)}$$
Solve each equation. If an equation has no solution, so indicate.$$\frac{2}{n-2}+\frac{1}{n+1}=\frac{1}{n^{2}-n-2}$$
Solve each equation. If an equation has no solution, so indicate.$$\frac{2 x+3}{x^{2}+5 x+6}+\frac{3 x-2}{x^{2}+x-6}=\frac{5 x-2}{x^{2}-4}$$
Solve each equation. If an equation has no solution, so indicate.$$\frac{3 x}{x^{2}+x}-\frac{2 x}{x^{2}+5 x}=\frac{x+2}{x^{2}+6 x+5}$$
Solve each equation. If an equation has no solution, so indicate.$$\frac{3 x+5}{x^{3}+8}+\frac{3}{x^{2}-4}=\frac{2(3 x-2)}{(x-2)\left(x^{2}-2 x+4\right)}$$
Solve each equation. If an equation has no solution, so indicate.$$\frac{1}{n+8}-\frac{3 n-4}{5 n^{2}+42 n+16}=\frac{1}{5 n+2}$$
Solve each equation. If an equation has no solution, so indicate.$$\frac{1}{11-n}-\frac{2(3 n-1)}{-7 n^{2}+74 n+33}=\frac{1}{7 n+3}$$
Solve each equation. If an equation has no solution, so indicate.$$\frac{4}{a^{2}-13 a-48}-\frac{2}{a^{2}-18 a+32}=\frac{1}{a^{2}+a-6}$$
Solve each equation. If an equation has no solution, so indicate.$$\frac{5}{y+4}+\frac{2}{y+2}=\frac{6}{y+2}-\frac{1}{y^{2}+6 y+8}$$
Solve each equation. If an equation has no solution, so indicate.$$\frac{6}{2 a-6}-\frac{3}{3-3 a}=\frac{1}{a^{2}-4 a+3}$$
Solve each equation. If an equation has no solution, so indicate.$$\frac{3 y}{6-3 y}+\frac{2 y}{2 y+4}=\frac{8}{4-y^{2}}$$
Solve each equation. If an equation has no solution, so indicate.$$\frac{3+2 a}{a^{2}+6+5 a}-\frac{2-3 a}{a^{2}-6+a}=\frac{5 a-2}{a^{2}-4}$$
Solve each equation. If an equation has no solution, so indicate.$$\frac{a}{a+2}-1=-\frac{3 a+2}{a^{2}+4 a+4}$$
Solve each equation. If an equation has no solution, so indicate.$$\frac{x-1}{x+3}+\frac{x-2}{x-3}=\frac{1-2 x}{3-x}$$
Solve each formula for the specified variable.$$k=2.2 p ; p$$
Solve each formula for the specified variable.$$a x+b=0 ; x$$
Solve each formula for the specified variable.$$P=2 l+2 w ; w$$
Solve each formula for the specified variable.$$V=\frac{1}{3} \pi r^{2} h ; h$$
Solve each formula for the specified variable.$$V=\frac{1}{3} \pi r^{2} h ; r^{2}$$
Solve each formula for the specified variable.$$z=\frac{x-\mu}{\sigma} ; \mu$$
Solve each formula for the specified variable.$$P_{n}=L+\frac{s i}{f} s$$
Solve each formula for the specified variable.$$P_{n}=L+\frac{s i}{f}, f$$
Solve each formula for the specified variable.$$F=\frac{m M g}{r^{2}}, m$$
Solve each formula for the specified variable.$$\frac{1}{f}=\frac{1}{p}+\frac{1}{q}, f$$
Solve each formula for the specified variable.$$\frac{x}{a}+\frac{y}{b}=1 ; y$$
Solve each formula for the specified variable.$$\frac{x}{a}-\frac{y}{b}=1 ; a$$
Solve each formula for the specified variable.$$\frac{1}{r}=\frac{1}{r_{1}}+\frac{1}{r_{2}} ; r$$
Solve each formula for the specified variable.$$\frac{1}{r}=\frac{1}{r_{1}}+\frac{1}{r_{2}}, r_{1}$$
Solve each formula for the specified variable.$$l=a+(n-1) d ; n$$
Solve each formula for the specified variable.$$l=a+(n-1) d ; d$$
Solve each formula for the specified variable.$$a=(n-2) \frac{180}{n}, n$$
Solve each formula for the specified variable.$$S=\frac{a-I r}{1-r} ; a$$
Solve each formula for the specified variable.$$R=\frac{1}{\frac{1}{r_{1}}+\frac{1}{r_{2}}+\frac{1}{r_{3}}} r_{1}$$
Solve each formula for the specified variable.$$R=\frac{1}{\frac{1}{r_{1}}+\frac{1}{r_{2}}+\frac{1}{r_{3}}}$$
Explain why a conditional linear equation always has exactly one root.
Define an extrancous solution and explain how such a solution occurs.
Simplify each expression. Use absolute value symbols when necessary.$$\left(25 x^{2}\right)^{12}$$
Simplify each expression. Use absolute value symbols when necessary.$$\left(\frac{25 p^{2}}{16 q^{4}}\right)^{1 / 2}$$
Simplify each expression. Use absolute value symbols when necessary.$$\left(\frac{125 x^{3}}{8 y^{6}}\right)^{-2 / 3}$$
Simplify each expression. Use absolute value symbols when necessary.$$\left(-\frac{27 y^{3}}{1,000 x^{6}}\right)^{1 / 3}$$
Simplify each expression. Use absolute value symbols when necessary.$$\sqrt{25 y^{2}}$$
Simplify each expression. Use absolute value symbols when necessary.$$\sqrt[3]{-125 y^{9}}$$
Simplify each expression. Use absolute value symbols when necessary.$$\sqrt[4]{\frac{a^{4} b^{12}}{z^{8}}}$$
Simplify each expression. Use absolute value symbols when necessary.$$\cdot \sqrt[5]{\frac{x^{10} y^{5}}{z^{15}}}$$