Verified answer

We have to consider two possible cases. This is generally the case when you have to solve equations or inequalities involving absolute values.

* Case 1: 8x ≥ 0:

This means your inequality becomes:

5x^2 - 14x - 3 ≤ 0

Solving the quadratic inequality gives: -1/5 ≤ x ≤ 3. Given that 8x ≥ 0, this means that x must be in [0,3].

* Case 2: 8x < 0:

Now we get:

5x^2 + 2x - 3 ≤ 0.

Solving the quadratic inequality gives:

-1 ≤ x ≤ 3/5. Given that 8x < 0, this means x must be in [-1,0).

Combining the two scenarios gives us the solution: x is in the range [-1, 3]. A graphical plot of the function confirms this.

Dear Kathy,

5x^(2)-6x-3<=|8x|

Since |8x| is on the right-hand side of the equation, switch the sides so it is on the left-hand side of the equation.

|8x|>=5x^(2)-6x-3

Remove the absolute value term. This creates a \ on the right-hand side of the equation because |x|=\x.

8x>=\(5x^(2)-6x-3)

Set up the + portion of the \ solution.

8x>=5x^(2)-6x-3

Solve the first equation for x.

-(1)/(5)<=x<=3

Set up the - portion of the \ solution. When solving the - portion of an inequality, flip the direction of the inequality sign.

8x<=-(5x^(2)-6x-3)

Since x is on the right-hand side of the equation, switch the sides so it is on the left-hand side of the equation.

-(5x^(2)-6x-3)>=8x

Multiply -1 by each term inside the parentheses.

-5x^(2)+6x+3>=8x

Move all terms not containing x to the right-hand side of the inequality.

-5x^(2)-2x+3>=0

Multiply each term in the equation by -1.

5x^(2)+2x-3<=0

Remove the fraction by multiplying the first term of the factor by the denominator of the second term.

(x+1)(5x-3)<=0

Set each of the factors of the left-hand side of the inequality equal to 0 to find the critical points.

x+1=0_5x-3=0

Since 1 does not contain the variable to solve for, move it to the right-hand side of the equation by subtracting 1 from both sides.

x=-1_5x-3=0

Set each of the factors of the left-hand side of the inequality equal to 0 to find the critical points.

x=-1_5x-3=0

Since -3 does not contain the variable to solve for, move it to the right-hand side of the equation by adding 3 to both sides.

x=-1_5x=3

Divide each term in the equation by 5.

x=-1_x=(3)/(5)

Since this is a 'less than 0' inequality, all intervals that make the expression negative are part of the solution.

-1<=x<=(3)/(5)

The solution to the inequality includes both the positive and negative versions of the absolute value.

-(1)/(5)<=x<=3 or -1<=x<=(3)/(5)

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