Question

Graphically Solve the given equation or inequality graphically 5x^{2}-x^{3}=-x^{2}+3x+4 and 5x^{2}-x^{3}\leq-x^{2}+3x+4

Decimals
ANSWERED
asked 2021-08-04
Solving Equations and Inequalities Graphically Solve the given equation or inequality graphically. State your answers rounded to two decimals.
a) \(\displaystyle{5}{x}^{{{2}}}-{x}^{{{3}}}=-{x}^{{{2}}}+{3}{x}+{4}\)
b) \(\displaystyle{5}{x}^{{{2}}}-{x}^{{{3}}}\leq-{x}^{{{2}}}+{3}{x}+{4}\)

Answers (1)

2021-08-05
Step 1
a) Definition: Solving Equations Graphically
The solution(s) of the equation \(\displaystyle{f{{\left({x}\right)}}}={g{{\left({x}\right)}}}\) are the values of x where the graphs of f and g intersect.
Step 2
Observation:
The provided equation is \(\displaystyle{5}{x}^{{{2}}}-{x}^{{{3}}}=-{x}^{{{2}}}+{3}{x}+{4}\)
From the definition, it can be directly consider that \(\displaystyle{f{{\left({x}\right)}}}={5}{x}^{{{2}}}-{x}^{{{3}}}\) and \(\displaystyle{\left({x}\right)}=-{x}^{{{2}}}+{3}{x}+{4}\)
By the definition of solving equation, find the intersection of functions f and g bby graphing it to get the solution.
Let's graph the function to get the intersection.
image
It can be observed from the above graph that the functions \(\displaystyle{f{{\left({x}\right)}}}\) and \(\displaystyle{g{{\left({x}\right)}}}\) are intersecting at the points \(\displaystyle{x}=-{0.58},\ {x}={1.29}\) and \(\displaystyle{x}={5.29}\)
Step 3
b) Definition: Solving Equations Graphically
The solution(s) of the inequality \(\displaystyle{f{{\left({x}\right)}}}\geq\ {g{{\left({x}\right)}}}\) are the values of x where the graphs of g is higher that the graph of f.
Interpretation:
The provided inequality is \(\displaystyle{5}{x}^{{{2}}}-{x}^{{{3}}}\leq-{x}^{{{2}}}+{3}{x}+{4}\)
From the definition, it can be directly consider that \(\displaystyle{f{{\left({x}\right)}}}={5}{x}^{{{2}}}-{x}^{{{3}}}\) and \(\displaystyle{\left({x}\right)}=-{x}^{{{2}}}+{3}{x}+{4}\)
By the definition of solving inequality, find the f lower than that of g.
It can be observed from graph of part (a) that the function \(\displaystyle{f{{\left({x}\right)}}}\) is lower than \(\displaystyle{g{{\left({x}\right)}}}\) between \(\displaystyle-{0.58}\) to \(\displaystyle{1.29}\) and between the range \(\displaystyle{5.29}\) to \(\displaystyle\infty\) Therefore, from the point \(\displaystyle{x}=-{0.58}\) to \(\displaystyle{x}={1.29}\) and from \(\displaystyle{x}={5.29}\) to \(\displaystyle{x}=\infty\) the function \(\displaystyle{f{{\left({x}\right)}}}\) is lower than or equal to \(\displaystyle{g{{\left({x}\right)}}}\).
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