4.08 Graphical solutions of equations and inequalities
Consider the graph of y = 2 x + 6. Using the graph, solve the inequality 2 x + 6 \geq 0.
Consider the graph of y = x - 6. Using the graph, solve the inequality x - 6 < 0.
Consider the graph of y = 5 x + 3. Using the graph, solve the inequality 5 x + 3 \leq 0.
Consider the graphs of y = x + 6 and
y = x - 7:
How many solutions does the inequality x + 6 \geq x - 7 have?
Consider the graph of the lines y = 3 and
y = 23 - 4 x:
Using the graphs, solve the inequality
23 - 4 x < 3.
Using the graphs, solve the inequality - 20 + 4 x \geq 0.
Consider the equation 2 \left(x - 1\right) - 3 = 7.
Solve for the value of x that satisfies the equation.
To verify the solution graphically, what two straight lines need to be graphed?
Graph these lines on the same number plane.
Hence find the value of x that satisfies the two equations.
Consider the inequality 2 x - 4 > 2 - 4 x.
Sketch the graphs of the lines for y = 2 x - 4 and y = 2 - 4 x.
Find the point of intersection of the lines.
Hence, solve the inequality 2 x - 4 > 2 - 4 x.
Consider the graph of the lines y = 17 and\\ y = 4 x - 3:
Using the graphs, solve the inequality
4 x - 3 < 17.
Consider the graphs of y = x + 5 and \\ y = 12 - x:
Using the graphs, solve the inequality
x + 5 > 12 - x.
Consider the function y = 2x - 6.
Sketch the graph of the line on a number plane.
Hence, solve the inequality 2x - 6 \gt 0.
Consider the function y = \dfrac{1}{2}x + 5.
Sketch the graph of the line on a number plane.
Hence, solve the inequality \dfrac{1}{2}x + 5\lt0
Consider the functions y = 2\left(x+1\right) \text{ and } y = - 3
Sketch both lines on the same number plane.
Hence, solve the inequality 2\left(x+1\right) \gt - 3
Consider the functions y = x + 4 \text{ and } y = 2x - 1
Sketch both lines on the same number plane.
Hence, solve the inequality x+4\lt2x-1
Use sign diagrams to solve the following inequalities:
Consider the graph of y = f \left( x \right):
Find the values of x for which f \left( x \right) = 0.
For what values of x is f \left( x \right) < 0?
For what values of x is f \left( x \right) > 0?
What is the x-coordinate of the vertex of f \left( x \right)?
Consider the graph of y = f \left( x \right):
For what values of x is f \left( x \right) < 0.
For what values of x is f \left( x \right) \geq 0?
How many real solutions are there to f \left( x \right) = 0?
Consider the function f \left( x \right) = 5 + 4 x - x^{2}:
Use the graph to solve the inequality
5 + 4 x - x^{2} > 0.
Consider the graph of y = f \left( x \right):
For what values of x is f \left( x \right) < 0?
For what value of x is f \left( x \right) \geq 0?
What is the axis of symmetry of f \left( x \right)?
What is the value of the discriminant of f \left( x \right)?
Consider the function f \left( x \right) = x^{2} - 4 x - 5.
Sketch the graph of the function.
Hence state the values of x for which f \left( x \right) \leq 0.
Consider the function f \left( x \right) = 3 x^{2} - 2 x - 8.
Solve the equation f \left( x \right) = 0.
Sketch the graph of the function.
Hence state the values of x for which f \left( x \right) \geq 0.
Consider the inequality \left(x - 3\right)^{2} \leq 0.
How many x-intercepts does the graph of y = \left(x - 3\right)^{2} have?
Solve the inequality.
Consider the function y = 2 x^{2} + 9 x + 8.
Determine the x-intercepts of the function.
Is the graph concave up or concave down?
Hence find the values of x for which y > 0.
Consider the function f \left( x \right) = x^{2} - 2 x.
Sketch the graph of the function.
Hence state the values of x for which f \left( x \right) \leq 8.
Consider the inequality x^{2} - 2 x \leq - x + 2.
Sketch the graphs of y = x^{2} - 2 x and y = - x + 2 on the same number plane.
State the x-values for the points of intersection.
Hence solve the inequality x^{2} - 2 x \leq - x + 2.
Consider the inequality x^{2} > 6 x - 5.
Sketch the graphs of y = x^{2} and y = 6 x - 5 on the same number plane.
State the x-values for the points of intersection.
Hence solve the inequality x^{2} > 6 x - 5.
Consider the inequality 3 x^{2} + x \geq 2 x^{2} + 2.
Sketch the graphs of y = 3 x^{2} + x and y = 2 x^{2} + 2 on the same number plane.
Hence or otherwise, solve the inequality 3 x^{2} + x \geq 2 x^{2} + 2.
Use sign diagrams to solve the following inequalities:
To solve the inequality x \leq 2 x - 3, Christa graphed y = x + 3. What other line would she need to graph to be able to solve the inequality graphically?
To solve the inequality x \leq \dfrac{x - 3}{4} - 1, Tracy graphed y = x - 3. What other line would she need to graph to be able to solve the inequality graphically?
Consider the graphs of y = 3 x + 4 and \\ y = x:
Using the graphs, solve the inequality \\ 3 x + 4 - x \leq 0.
Consider the graph of y = - \dfrac{2}{3} x - 2.
Using the graphs, solve the inequality
- 2 x - 6 > 0.
Consider the functions y = 2x+1 \text{ and } y = 4 - x.
Sketch both lines on the same number plane.
Hence, solve the inequality 2x + 1 \lt 4 - x.
Consider the functions y = 3x-3 \text{ and } y = 0.
Sketch both lines on the same number plane.
Hence, solve the inequality 3x - 3\lt 0.
Explain why the solutions to the above inequalities are the same.
Explain whether the following inequalities can be solved graphically using the graphs of: f(x) = 4 x + 3 \text{ and }g(x) = 5 - x
\left( 4 x + 3\right) - \left(5 - x\right) \leq 0
\left( 4 x + 3\right) + \left(3 - x\right) < 0
\dfrac{4 x + 3}{5} + x > 0
3 x + 8 < 0
5 x > 2