# MATH 1450 Chapter 1 Review

**1 Linear Equations
Tool:** We can add, subtract, multiply and divide both sides of

an equation by the same quantity (except no division by

zero). So we can simplify equations to solve for variables.

If a=b , then (assuming no zeros in denominators )

a+ c=b+ c

a-c=b-c

a· c=b· c

a/c=b/c

1/a=1/b

**Tool:**Absolute value equations may be split into two linear

equations without absolute value bars . The equation

∣ax +b∣=k

means

ax+ b=k or ax+ b=-k

**Tool:**In word problems there is no recipe. Look at the

question to find the variable(s) you need. Put together with

any given (or implied) numbers and see if the equation makes

sense. Look at the different types of word problems in the

text to gain a feel for what’s out there.

2 Nonlinear equations

2 Nonlinear equations

**Tool:**Group terms on one side of equation and factor. If

(x-r) (x-s)=0 , then x=r or x=s

**Tool:**Isolate square on one side and take square root of both

sides. The equation (ax+b)

^{2}=k is equivalent to the pair

of equations

**Tool:**Completing the square (see p. 54 for alternate version).

Use for quadratic equations ax

^{2}+bx+c=0 too hard to

factor. Subtract c from both sides and divide through by a

to get . Then add to both sides and

factor to obtain , which you can square root .

**Tool:**Quadratic formula gives equivalent solution to

completing the square. Use when coefficients are nasty .

For ax

^{2}+bx+c=0 we have two solutions

**Basic Equation: (solve for s)**

** Rational Equation : (solve for r, check extraneous
solution)**

**Formulas: (solve for h)**

**Absolute Value Equations: (solve for x)**

**Linear Word problem:**

A gambler plays the stock market. He bets $10000 spread

between a low risk stock returning 5% and a high risk stock

returning 10%. Assuming no crashes, total income for the

year is $850. How much did he invest in each stock?

**Factor problem: (solve for w)
**w=w

^{2}+12

**Square root problem : (solve for y)**

Recall that .

(3y+4 )

^{2}+4=0

**Completing the square problem: (solve for x)**

x

^{2}-4x+5=0

**Polynomial factoring : (solve for x, four solutions)**

x

^{4}-16=0

**Rational quadratic: (solve for v, check extraneous solution)**

Quadratic formula problem : (solve for t)

Quadratic formula problem : (solve for t)

1.21t

^{2}-3.54t+7.29=0

Ans: t=1.463±1.971i

**Tool:** Solving equations with roots you solve by
raising both

sides to the appropriate power and simplifying/solving.

Recall that and
.

Note that in some problems you may have to square both

sides more than once to remove the roots .

**3 Inequalities**

**Tool: **Just like for equations, we can add, subtract, multiply

and divide both sides of an inequality by the same quantity

(except no division by zero). The only catch is that if you

multiply or divide by a negative quantity, you must switch

the sign (this means no multiplying through by variables,

because you don't know if they are positive or negative ).

**Tool:** Absolute value inequalities may be transformed into

linear inequalities as follows . (It helps to think of absolute

value as denoting size or distance.) If ∣x∣< k , then x is no

larger than k in magnitude and -k < x< k .

Alternatively, if we switch to ∣x∣>k , then x lies outside

of a bounded interval, so x< -k or x>k .

**Tool:** Nonlinear inequalities that are factorable may use the

cut- point method (p. 68). In summary you look at each

factor and its zero on the number line. Decide whether each

zero is a solution. The zeros mark out intervals on the

number line. Pick a test point on the interior of each interval.

If the test point solves the inequality, mark the interval as part

of the solution.

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