Practice Linear Inequalities Problems, learn about the domain & the range of a relation, linear function and find the domain of the expression.
Consider a linear function where \(a>0\), \(f(x)=ax+b\), suppose we want to know for what values of \(x\) is the expression positive, i.e.
\(ax+b>0\rightarrow\) this is called a linear inequality or linear equation.
To solve this consider:
\(ax+b>0\) |
Subtract \(-b\) from both sides |
\(ax>-b\) |
Divide by \('a'\) on both sides |
\(x>\dfrac {-b}{a}\) |
Solution to linear inequality, also represented by \(x\in\left (\dfrac {-b}{a},\infty \right)\) |
Similar step for solving \(ax+b<0\) |
In general:
For representing values of \(x\) which are between \('a'\) and \('b'\) excluding them
\(a\leq x\leq b\)is represented by \(x\in[a, b]\rightarrow\) Closed interval
\(a\leq x< b\)is represented by \(x\in[a, b)\rightarrow\) Half open interval
A \(x<1\)
B \(x>\dfrac {7}{2}\)
C \(x<-2\)
D \(x=0\)
\(ax + b \geq0 \space or\, \space ax + b \leq 0\)
A \(x \leq \Large {7 \over 3}\)
B \(x \geq 2\)
C \(x \leq 10\)
D \(x \geq -1\)
\(f(\underbrace{x}_{domain})=\underbrace{ formula \,of\, an\,expression}_{range}\)
A \(x \epsilon \space (-\infty, 1/9]\)
B \(x \epsilon \space [1, \infty)\)
C \(x \epsilon \space [1/7, \infty)\)
D \(x \epsilon \space (-\infty, 8]\)
A \(\{x \space | \space x \neq -3, x \neq 2 \}\)
B \(\{x \space | \space x \neq 10, x \neq 1 \}\)
C \(\{x \space | \space x \neq 5, x \neq 7 \}\)
D \(\{x \space | \space x \neq -5, x \neq -10 \}\)
A \(\left \{ x\mid x\,\neq \dfrac {2}{3} \right \}\)
B \(\left \{ x\mid x\,\neq \dfrac {3}{2} \right \}\)
C \(\left \{ x\mid x\,\neq 5 \right \}\)
D \(\left \{ x\mid x\,\neq 2 \right \}\)