Substitute our variables into the equation for the

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substitute our variables into the equation for the perimeter. The formula already calls for w , so we can just leave that as is. Where it calls for length, we can just plug in “2 w + 3” . We already know our perimeter is 48ft, so we substitute that in for p . Use the distributive property to multiply 2 by 2 w and 3. Combine like terms (4 w and 2 w ). Subtract 6 from both sides to “undo” the addition. Divide by 6 on both sides to “undo” t he multiplication. You are left with a width of 7 ft. Now we need to find the length. w
Check to see that it works 2(7) + 3 14 + 3 17ft = length Your turn: I have a box. The length of the box is 12 in. The height of the box is 5 in. The box has a total volume of 360 in. 2 What is the width of the box? Note: The formula for volume is V = lwh , where v = volume, l = length, w = width, and h = height. (You should get w = 6in.) Inequalities We’ve all been taught little tricks to remember the inequality sign. For example; when given x < 10, we know that x is less than ten because x has the LITTLE side of the sign and 10 has the BIG side of the sign. Solving inequalities is similar to solving equations; what you do to one side of an inequality, we must do to the other. If we are given x + 7 > 13 and asked to solve, we would undo the addition on the left side by subtracting 7 from both sides. We would then be left with x > 6 , which is our answer. Suppose we were given ¼ x < 2. To undo the division, we would multiply both sides by 4. The result would be x < 8 . But if we had ¼ x < 2 , we would multiply by both sided of the inequality by -4 and the rule is that when multiplying (or dividing) by a negative number , we must always flip the sign of an inequality . So we would get x > 2. Substitute our newly found width and simplify using order of operations So now we know that the yard is 7ft wide and 17ft long.
Let’s practice: It’s always good to check our answer. To do this, plug values in for x. Let’s try 1 , since it is greater than 0 and an easier number to work with. 1 5 is - 4. -4 times -2 is -8. Is -8 < 10? You bet! Your turn: 4x + 2 > 10 [You should get x > 2] Absolute Values The absolute value of a number is its distance from zero on the number line. Since we can’t have negative distance, absolute values are always positive. │5│= 5 - 7│= 7 - 8 + 2│= 6 When solving absolute value equations, we must consider that the number inside could have been negative before you applied the absolute value! For example: │x + 2│= 3 x + 2 and x + 2 3 -3 When we take the absolute value, both are equal to 3. So we end up with TWO SOLUTIONS for this equation. Solve: -2(x 5) < 10 -2 -2 x 5 > -5 + 5 + 5 x > 0 Just as with an equation, we start here by dividing both sides by -2. Since we are dividing by a negative, we must flip the inequality sign. Next, we add five to both sides and we are left with x > 0.
x + 2 = 3 and x + 2 = -3 -1 -1 -2 -2 x = 1 x = -5 Your turn: │3x – 4│= 5 [You should get x = -1/3 and x = 3] Linear Equations If we plot all the solutions to a linear equation on a graph, they form a line. That is why they are called line ar equations . Linear equations can be written in slope-intercept form . There are two important things to know when writing the equation of a line in slope-intercept form
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